Loogle!

Result

Found 243 declarations mentioning Set.PairwiseDisjoint. Of these, only the first 200 are shown.

• Set.PairwiseDisjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] (s : Set ι) (f : ι → α) : Prop
• Set.pairwiseDisjoint_empty 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {f : ι → α} : ∅.PairwiseDisjoint f
• Set.pairwiseDisjoint_singleton 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] (i : ι) (f : ι → α) : {i}.PairwiseDisjoint f
• Pairwise.pairwiseDisjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {f : ι → α} (h : Pairwise (Function.onFun Disjoint f)) (s : Set ι) : s.PairwiseDisjoint f
• Set.PairwiseDisjoint.subset 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s t : Set ι} {f : ι → α} (ht : t.PairwiseDisjoint f) (h : s ⊆ t) : s.PairwiseDisjoint f
• Set.pairwiseDisjoint_singleton' 📋 Mathlib.Data.Set.Pairwise.Basic
  {ι : Type u_4} (s : Set ι) : s.PairwiseDisjoint singleton
• Set.PairwiseDisjoint.mono 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f g : ι → α} (hs : s.PairwiseDisjoint f) (h : g ≤ f) : s.PairwiseDisjoint g
• Set.InjOn.pairwiseDisjoint_image 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [PartialOrder α] [OrderBot α] {f : ι → α} {g : ι' → ι} {s : Set ι'} (h : Set.InjOn g s) : (g '' s).PairwiseDisjoint f ↔ s.PairwiseDisjoint (f ∘ g)
• Set.pairwiseDisjoint_range_singleton 📋 Mathlib.Data.Set.Pairwise.Basic
  {ι : Type u_4} : (Set.range singleton).PairwiseDisjoint id
• Set.PairwiseDisjoint.mono_on 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f g : ι → α} (hs : s.PairwiseDisjoint f) (h : ∀ ⦃i : ι⦄, i ∈ s → g i ≤ f i) : s.PairwiseDisjoint g
• Set.PairwiseDisjoint.elim 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (h : ¬Disjoint (f i) (f j)) : i = j
• Set.PairwiseDisjoint.eq_or_disjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (h : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) : i = j ∨ Disjoint (f i) (f j)
• Set.pairwiseDisjoint_fiber 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} (f : ι → α) (s : Set α) : s.PairwiseDisjoint fun a => f ⁻¹' {a}
• Set.PairwiseDisjoint.image_of_le 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {g : ι → ι} (hg : f ∘ g ≤ f) : (g '' s).PairwiseDisjoint f
• Set.pairwiseDisjoint_singleton_iff_injOn 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {s : Set ι} {f : ι → α} : (s.PairwiseDisjoint fun i => {f i}) ↔ Set.InjOn f s
• Set.PairwiseDisjoint.insert 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {i : ι} (h : ∀ j ∈ s, i ≠ j → Disjoint (f i) (f j)) : (insert i s).PairwiseDisjoint f
• Set.pairwiseDisjoint_insert 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} {i : ι} : (insert i s).PairwiseDisjoint f ↔ s.PairwiseDisjoint f ∧ ∀ j ∈ s, i ≠ j → Disjoint (f i) (f j)
• Set.PairwiseDisjoint.insert_of_notMem 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {i : ι} (hi : i ∉ s) (h : ∀ j ∈ s, Disjoint (f i) (f j)) : (insert i s).PairwiseDisjoint f
• Set.PairwiseDisjoint.range 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} (g : ↑s → ι) (hg : ∀ (i : ↑s), f (g i) ≤ f ↑i) (ht : s.PairwiseDisjoint f) : (Set.range g).PairwiseDisjoint f
• Set.pairwiseDisjoint_insert_of_notMem 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s : Set ι} {f : ι → α} {i : ι} (hi : i ∉ s) : (insert i s).PairwiseDisjoint f ↔ s.PairwiseDisjoint f ∧ ∀ j ∈ s, Disjoint (f i) (f j)
• Set.PairwiseDisjoint.elim' 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [SemilatticeInf α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (h : f i ⊓ f j ≠ ⊥) : i = j
• Set.PairwiseDisjoint.union 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s t : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) (ht : t.PairwiseDisjoint f) (h : ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ t → i ≠ j → Disjoint (f i) (f j)) : (s ∪ t).PairwiseDisjoint f
• Set.pairwiseDisjoint_union 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {s t : Set ι} {f : ι → α} : (s ∪ t).PairwiseDisjoint f ↔ s.PairwiseDisjoint f ∧ t.PairwiseDisjoint f ∧ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ t → i ≠ j → Disjoint (f i) (f j)
• Set.PairwiseDisjoint.eq_of_le 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} [SemilatticeInf α] [OrderBot α] {s : Set ι} {f : ι → α} (hs : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (hf : f i ≠ ⊥) (hij : f i ≤ f j) : i = j
• Set.PairwiseDisjoint.elim_set 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {s : Set ι} {f : ι → Set α} (hs : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (a : α) (hai : a ∈ f i) (haj : a ∈ f j) : i = j
• Set.exists_ne_mem_inter_of_not_pairwiseDisjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {s : Set ι} {f : ι → Set α} (h : ¬s.PairwiseDisjoint f) : ∃ i ∈ s, ∃ j ∈ s, i ≠ j ∧ ∃ x, x ∈ f i ∩ f j
• Set.pairwiseDisjoint_range_iff 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_6} {β : Type u_7} {f : α → Set β} : (Set.range f).PairwiseDisjoint id ↔ ∀ (x y : α), f x ≠ f y → Disjoint (f x) (f y)
• Set.pairwiseDisjoint_image_right_iff 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {s : Set α} {t : Set β} (hf : ∀ a ∈ s, Function.Injective (f a)) : (s.PairwiseDisjoint fun a => f a '' t) ↔ Set.InjOn (fun p => f p.1 p.2) (s ×ˢ t)
• Set.pairwiseDisjoint_image_left_iff 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {s : Set α} {t : Set β} (hf : ∀ b ∈ t, Function.Injective fun a => f a b) : (t.PairwiseDisjoint fun b => (fun a => f a b) '' s) ↔ Set.InjOn (fun p => f p.1 p.2) (s ×ˢ t)
• Set.exists_lt_mem_inter_of_not_pairwiseDisjoint 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {ι : Type u_4} {s : Set ι} [LinearOrder ι] {f : ι → Set α} (h : ¬s.PairwiseDisjoint f) : ∃ i ∈ s, ∃ j ∈ s, i < j ∧ ∃ x, x ∈ f i ∩ f j
• Set.pairwiseDisjoint_pi 📋 Mathlib.Data.Set.Pairwise.Basic
  {ι : Type u_4} {ι' : ι → Type u_6} {α : ι → Type u_7} {s : (i : ι) → Set (ι' i)} {f : (i : ι) → ι' i → Set (α i)} (hs : ∀ (i : ι), (s i).PairwiseDisjoint (f i)) : (Set.univ.pi s).PairwiseDisjoint fun I => Set.univ.pi fun i => f i (I i)
• Set.PairwiseDisjoint.prod 📋 Mathlib.Data.Set.Pairwise.Basic
  {α : Type u_1} {β : Type u_2} {ι : Type u_4} {ι' : Type u_5} {s : Set ι} {t : Set ι'} {f : ι → Set α} {g : ι' → Set β} (hs : s.PairwiseDisjoint f) (ht : t.PairwiseDisjoint g) : (s ×ˢ t).PairwiseDisjoint fun i => f i.1 ×ˢ g i.2
• Finset.pairwiseDisjoint_singleton_iff_injOn 📋 Mathlib.Data.Finset.Disjoint
  {ι : Type u_1} {α : Type u_2} {s : Set ι} {f : ι → α} : (s.PairwiseDisjoint fun i => {f i}) ↔ Set.InjOn f s
• Finset.pairwiseDisjoint_coe 📋 Mathlib.Data.Finset.Disjoint
  {α : Type u_2} {ι : Type u_5} {s : Set ι} {f : ι → Finset α} : (s.PairwiseDisjoint fun i => ↑(f i)) ↔ s.PairwiseDisjoint f
• Set.pairwiseDisjoint_filter 📋 Mathlib.Data.Finset.Filter
  {α : Type u_1} {β : Type u_2} [DecidableEq β] (f : α → β) (s : Set β) (t : Finset α) : s.PairwiseDisjoint fun x => {x_1 ∈ t | f x_1 = x}
• Finset.pairwiseDisjoint_iff 📋 Mathlib.Data.Finset.Basic
  {α : Type u_1} [DecidableEq α] {ι : Type u_3} {s : Set ι} {f : ι → Finset α} : s.PairwiseDisjoint f ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f i ∩ f j).Nonempty → i = j
• Finset.disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (s : Finset α) (t : α → Finset β) (hf : (↑s).PairwiseDisjoint t) : Finset β
• Finset.disjiUnion_eq_biUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} [DecidableEq β] (s : Finset α) (f : α → Finset β) (hf : (↑s).PairwiseDisjoint f) : s.disjiUnion f hf = s.biUnion f
• Finset.disjiUnion_val 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (s : Finset α) (t : α → Finset β) (h : (↑s).PairwiseDisjoint t) : (s.disjiUnion t h).val = s.val.bind fun a => (t a).val
• Finset.singleton_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {t : α → Finset β} (a : α) {h : (↑{a}).PairwiseDisjoint t} : {a}.disjiUnion t h = t a
• Finset.pairwiseDisjoint_filter 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {s : Finset α} {f : α → Finset β} (h : (↑s).PairwiseDisjoint f) (p : β → Prop) [DecidablePred p] : (↑s).PairwiseDisjoint fun a => Finset.filter p (f a)
• Finset.filter_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (s : Finset α) (f : α → Finset β) (h : (↑s).PairwiseDisjoint f) (p : β → Prop) [DecidablePred p] : Finset.filter p (s.disjiUnion f h) = s.disjiUnion (fun a => Finset.filter p (f a)) ⋯
• Finset.disjoint_disjiUnion_left 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (s : Finset α) (f : α → Finset β) (hf : (↑s).PairwiseDisjoint f) (t : Finset β) : Disjoint (s.disjiUnion f hf) t ↔ ∀ i ∈ s, Disjoint (f i) t
• Finset.disjoint_disjiUnion_right 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (s : Finset β) (t : Finset α) (f : α → Finset β) (hf : (↑t).PairwiseDisjoint f) : Disjoint s (t.disjiUnion f hf) ↔ ∀ i ∈ t, Disjoint s (f i)
• Finset.mem_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {s : Finset α} {t : α → Finset β} {b : β} {h : (↑s).PairwiseDisjoint t} : b ∈ s.disjiUnion t h ↔ ∃ a ∈ s, b ∈ t a
• Finset.coe_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {s : Finset α} {t : α → Finset β} {h : (↑s).PairwiseDisjoint t} : ↑(s.disjiUnion t h) = ⋃ x ∈ ↑s, ↑(t x)
• Finset.fold_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {f : α → β} {op : β → β → β} [hc : Std.Commutative op] [ha : Std.Associative op] {ι : Type u_4} {s : Finset ι} {t : ι → Finset α} {b : ι → β} {b₀ : β} (h : (↑s).PairwiseDisjoint t) : Finset.fold op (Finset.fold op b₀ b s) f (s.disjiUnion t h) = Finset.fold op b₀ (fun i => Finset.fold op (b i) f (t i)) s
• Finset.sUnion_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {f : α → Finset (Set β)} (I : Finset α) (hf : (↑I).PairwiseDisjoint f) : ⋃₀ ↑(I.disjiUnion f hf) = ⋃ a ∈ I, ⋃₀ ↑(f a)
• Finset.pairwiseDisjoint_disjUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {s : Finset α} {f g : α → Finset β} (hfg : ∀ (a : α), Disjoint (f a) (g a)) (hfg' : (↑s).Pairwise fun a₁ a₂ => Disjoint (f a₁) (g a₂)) (hf : (↑s).PairwiseDisjoint f) (hg : (↑s).PairwiseDisjoint g) : (↑s).PairwiseDisjoint fun a => (f a).disjUnion (g a) ⋯
• Finset.disjiUnion_disjUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {s : Finset α} {f g : α → Finset β} (hfg : ∀ (a : α), Disjoint (f a) (g a)) (hfg' : (↑s).Pairwise fun a₁ a₂ => Disjoint (f a₁) (g a₂)) (hf : (↑s).PairwiseDisjoint f) (hg : (↑s).PairwiseDisjoint g) : s.disjiUnion (fun a => (f a).disjUnion (g a) ⋯) ⋯ = (s.disjiUnion f hf).disjUnion (s.disjiUnion g hg) ⋯
• Finset.disjiUnion_map 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Finset α} {t : α → Finset β} {f : β ↪ γ} {h : (↑s).PairwiseDisjoint t} : Finset.map f (s.disjiUnion t h) = s.disjiUnion (fun a => Finset.map f (t a)) ⋯
• Finset.map_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α ↪ β} {s : Finset α} {t : β → Finset γ} {h : (↑(Finset.map f s)).PairwiseDisjoint t} : (Finset.map f s).disjiUnion t h = s.disjiUnion (fun a => t (f a)) ⋯
• Finset.disjiUnion_cons 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} (a : α) (s : Finset α) (ha : a ∉ s) (f : α → Finset β) (H : (↑(Finset.cons a s ha)).PairwiseDisjoint f) : (Finset.cons a s ha).disjiUnion f H = (f a).disjUnion (s.disjiUnion f ⋯) ⋯
• Finset.disjiUnion_disjiUnion 📋 Mathlib.Data.Finset.Union
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (s : Finset α) (f : α → Finset β) (g : β → Finset γ) (h1 : (↑s).PairwiseDisjoint f) (h2 : (↑(s.disjiUnion f h1)).PairwiseDisjoint g) : (s.disjiUnion f h1).disjiUnion g h2 = s.attach.disjiUnion (fun a => (f ↑a).disjiUnion g ⋯) ⋯
• Set.pairwiseDisjoint_smul_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} [SMul α β] [IsLeftCancelSMul α β] {s : Set α} {t : Set β} : (s.PairwiseDisjoint fun x => x • t) ↔ Set.InjOn (fun p => p.1 • p.2) (s ×ˢ t)
• Set.pairwiseDisjoint_vadd_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} [VAdd α β] [IsLeftCancelVAdd α β] {s : Set α} {t : Set β} : (s.PairwiseDisjoint fun x => x +ᵥ t) ↔ Set.InjOn (fun p => p.1 +ᵥ p.2) (s ×ˢ t)
• Set.pairwiseDisjoint_iUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [PartialOrder α] [OrderBot α] {f : ι → α} {g : ι' → Set ι} (h : Directed (fun x1 x2 => x1 ⊆ x2) g) : (⋃ n, g n).PairwiseDisjoint f ↔ ∀ ⦃n : ι'⦄, (g n).PairwiseDisjoint f
• Set.pairwiseDisjoint_sUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} [PartialOrder α] [OrderBot α] {f : ι → α} {s : Set (Set ι)} (h : DirectedOn (fun x1 x2 => x1 ⊆ x2) s) : (⋃₀ s).PairwiseDisjoint f ↔ ∀ ⦃a : Set ι⦄, a ∈ s → a.PairwiseDisjoint f
• Set.pairwiseDisjoint_iff 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {f : ι → Set α} {s : Set ι} : s.PairwiseDisjoint f ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → (f i ∩ f j).Nonempty → i = j
• Set.biUnionEqSigmaOfDisjoint 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {s : Set ι} {f : ι → Set α} (h : s.PairwiseDisjoint f) : ↑(⋃ i ∈ s, f i) ≃ (i : ↑s) × ↑(f ↑i)
• Set.pairwiseDisjoint_pair_insert 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {s : Set α} {a : α} (ha : a ∉ s) : (𝒫 s).PairwiseDisjoint fun t => {t, insert a t}
• pairwiseDisjoint_unique 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {f : ι → Set α} {s : Set ι} {y : α} (h_disjoint : s.PairwiseDisjoint f) (hy : y ∈ ⋃ i ∈ s, f i) : ∃! i, i ∈ s ∧ y ∈ f i
• Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {f : ι → Set α} {s t : Set ι} (h₀ : (s ∪ t).PairwiseDisjoint f) (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t
• Set.biUnion_diff_biUnion_eq 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f) : (⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i = ⋃ i ∈ s \ t, f i
• Set.biUnion_sdiff_biUnion_eq 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f) : (⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i = ⋃ i ∈ s \ t, f i
• Set.PairwiseDisjoint.biUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [CompleteLattice α] {s : Set ι'} {g : ι' → Set ι} {f : ι → α} (hs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i) (hg : ∀ i ∈ s, (g i).PairwiseDisjoint f) : (⋃ i ∈ s, g i).PairwiseDisjoint f
• Set.pairwiseDisjoint_prod_left 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [Order.Frame α] {s : Set ι} {t : Set ι'} {f : ι × ι' → α} : (s ×ˢ t).PairwiseDisjoint f ↔ (s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')) ∧ t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
• Set.PairwiseDisjoint.prod_left 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [CompleteLattice α] {s : Set ι} {t : Set ι'} {f : ι × ι' → α} (hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')) (ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')) : (s ×ˢ t).PairwiseDisjoint f
• Set.coe_biUnionEqSigmaOfDisjoint_symm_apply 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_5} {ι : Type u_6} {s : Set ι} {f : ι → Set α} (h : s.PairwiseDisjoint f) (x : (i : ↑s) × ↑(f ↑i)) : ↑((Set.biUnionEqSigmaOfDisjoint h).symm x) = ↑x.snd
• Set.coe_snd_biUnionEqSigmaOfDisjoint 📋 Mathlib.Data.Set.Pairwise.Lattice
  {α : Type u_5} {ι : Type u_6} {s : Set ι} {f : ι → Set α} (h : s.PairwiseDisjoint f) (x : ↑(⋃ i ∈ s, f i)) : ↑((Set.biUnionEqSigmaOfDisjoint h) x).snd = ↑x
• Finset.pairwiseDisjoint_pair_insert 📋 Mathlib.Data.Finset.Powerset
  {α : Type u_1} {s : Finset α} [DecidableEq α] {a : α} (ha : a ∉ s) : (↑s.powerset).PairwiseDisjoint fun t => {t, insert a t}
• Set.PairwiseDisjoint.finite_biUnion_iff 📋 Mathlib.Data.Set.Finite.Lattice
  {α : Type u} {β : Type v} {f : β → Set α} {s : Set β} (hs : s.PairwiseDisjoint f) : (⋃ i ∈ s, f i).Finite ↔ (∀ i ∈ s, (f i).Finite) ∧ {i | i ∈ s ∧ (f i).Nonempty}.Finite
• Finset.pairwiseDisjoint_map_sigmaMk 📋 Mathlib.Data.Finset.Sigma
  {ι : Type u_1} {α : ι → Type u_2} {s : Finset ι} {t : (i : ι) → Finset (α i)} : (↑s).PairwiseDisjoint fun i => Finset.map (Function.Embedding.sigmaMk i) (t i)
• Finset.card_disjiUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {M : Type u_4} (s : Finset ι) (t : ι → Finset M) (h : (↑s).PairwiseDisjoint t) : (s.disjiUnion t h).card = ∑ a ∈ s, (t a).card
• Finset.card_biUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {M : Type u_4} {s : Finset ι} [DecidableEq M] {t : ι → Finset M} (h : (↑s).PairwiseDisjoint t) : (s.biUnion t).card = ∑ u ∈ s, (t u).card
• Finset.prod_disjiUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [CommMonoid M] {f : ι → M} (s : Finset κ) (t : κ → Finset ι) (h : (↑s).PairwiseDisjoint t) : ∏ x ∈ s.disjiUnion t h, f x = ∏ i ∈ s, ∏ x ∈ t i, f x
• Finset.sum_disjiUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [AddCommMonoid M] {f : ι → M} (s : Finset κ) (t : κ → Finset ι) (h : (↑s).PairwiseDisjoint t) : ∑ x ∈ s.disjiUnion t h, f x = ∑ i ∈ s, ∑ x ∈ t i, f x
• Finset.prod_biUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [CommMonoid M] {f : ι → M} [DecidableEq ι] {s : Finset κ} {t : κ → Finset ι} (hs : (↑s).PairwiseDisjoint t) : ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i
• Finset.sum_biUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [AddCommMonoid M] {f : ι → M} [DecidableEq ι] {s : Finset κ} {t : κ → Finset ι} (hs : (↑s).PairwiseDisjoint t) : ∑ x ∈ s.biUnion t, f x = ∑ x ∈ s, ∑ i ∈ t x, f i
• Finset.prod_image_of_disjoint 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [CommMonoid M] [DecidableEq ι] [PartialOrder ι] [OrderBot ι] {f : κ → ι} {g : ι → M} (hg_bot : g ⊥ = 1) {I : Finset κ} (hf_disj : (↑I).PairwiseDisjoint f) : ∏ s ∈ Finset.image f I, g s = ∏ i ∈ I, g (f i)
• Finset.sum_image_of_disjoint 📋 Mathlib.Algebra.BigOperators.Group.Finset.Basic
  {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [AddCommMonoid M] [DecidableEq ι] [PartialOrder ι] [OrderBot ι] {f : κ → ι} {g : ι → M} (hg_bot : g ⊥ = 0) {I : Finset κ} (hf_disj : (↑I).PairwiseDisjoint f) : ∑ s ∈ Finset.image f I, g s = ∑ i ∈ I, g (f i)
• Set.PairwiseDisjoint.sSupIndep 📋 Mathlib.Order.SupIndep
  {α : Type u_1} [Order.Frame α] {s : Set α} : s.PairwiseDisjoint id → sSupIndep s
• sSupIndep_iff_pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} [Order.Frame α] {s : Set α} : sSupIndep s ↔ s.PairwiseDisjoint id
• sSupIndep.pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : sSupIndep s) : s.PairwiseDisjoint id
• Finset.SupIndep.pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} (hs : s.SupIndep f) : (↑s).PairwiseDisjoint f
• Set.PairwiseDisjoint.supIndep 📋 Mathlib.Order.SupIndep
  {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : (↑s).PairwiseDisjoint f → s.SupIndep f
• Finset.supIndep_iff_pairwiseDisjoint 📋 Mathlib.Order.SupIndep
  {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : s.SupIndep f ↔ (↑s).PairwiseDisjoint f
• Finset.card_le_card_biUnion 📋 Mathlib.Algebra.Order.BigOperators.Group.Finset
  {ι : Type u_1} {α : Type u_2} [DecidableEq α] {s : Finset ι} {f : ι → Finset α} (hs : (↑s).PairwiseDisjoint f) (hf : ∀ i ∈ s, (f i).Nonempty) : s.card ≤ (s.biUnion f).card
• Finset.card_le_card_biUnion_add_one 📋 Mathlib.Algebra.Order.BigOperators.Group.Finset
  {ι : Type u_1} {α : Type u_2} [DecidableEq α] {s : Finset ι} {f : ι → Finset α} (hf : Function.Injective f) (hs : (↑s).PairwiseDisjoint f) : s.card ≤ (s.biUnion f).card + 1
• Finset.card_le_card_biUnion_add_card_fiber 📋 Mathlib.Algebra.Order.BigOperators.Group.Finset
  {ι : Type u_1} {α : Type u_2} [DecidableEq α] {s : Finset ι} {f : ι → Finset α} (hs : (↑s).PairwiseDisjoint f) : s.card ≤ (s.biUnion f).card + {i ∈ s | f i = ∅}.card
• finprod_mem_sUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {M : Type u_5} [CommMonoid M] {f : α → M} {t : Set (Set α)} (h : t.PairwiseDisjoint id) (ht₀ : t.Finite) (ht₁ : ∀ x ∈ t, x.Finite) : ∏ᶠ (a : α) (_ : a ∈ ⋃₀ t), f a = ∏ᶠ (s : Set α) (_ : s ∈ t) (a : α) (_ : a ∈ s), f a
• finsum_mem_sUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {M : Type u_5} [AddCommMonoid M] {f : α → M} {t : Set (Set α)} (h : t.PairwiseDisjoint id) (ht₀ : t.Finite) (ht₁ : ∀ x ∈ t, x.Finite) : ∑ᶠ (a : α) (_ : a ∈ ⋃₀ t), f a = ∑ᶠ (s : Set α) (_ : s ∈ t) (a : α) (_ : a ∈ s), f a
• finprod_mem_biUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {ι : Type u_3} {M : Type u_5} [CommMonoid M] {f : α → M} {I : Set ι} {t : ι → Set α} (h : I.PairwiseDisjoint t) (hI : I.Finite) (ht : ∀ i ∈ I, (t i).Finite) : ∏ᶠ (a : α) (_ : a ∈ ⋃ x ∈ I, t x), f a = ∏ᶠ (i : ι) (_ : i ∈ I) (j : α) (_ : j ∈ t i), f j
• finsum_mem_biUnion 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {ι : Type u_3} {M : Type u_5} [AddCommMonoid M] {f : α → M} {I : Set ι} {t : ι → Set α} (h : I.PairwiseDisjoint t) (hI : I.Finite) (ht : ∀ i ∈ I, (t i).Finite) : ∑ᶠ (a : α) (_ : a ∈ ⋃ x ∈ I, t x), f a = ∑ᶠ (i : ι) (_ : i ∈ I) (j : α) (_ : j ∈ t i), f j
• Finset.card_dvd_card_image₂_right 📋 Mathlib.Data.Finset.NAry
  {α : Type u_1} {β : Type u_3} {γ : Type u_5} [DecidableEq γ] {f : α → β → γ} {s : Finset α} {t : Finset β} (hf : ∀ a ∈ s, Function.Injective (f a)) (hs : ((fun a => Finset.image (f a) t) '' ↑s).PairwiseDisjoint id) : t.card ∣ (Finset.image₂ f s t).card
• Finset.card_dvd_card_image₂_left 📋 Mathlib.Data.Finset.NAry
  {α : Type u_1} {β : Type u_3} {γ : Type u_5} [DecidableEq γ] {f : α → β → γ} {s : Finset α} {t : Finset β} (hf : ∀ b ∈ t, Function.Injective fun a => f a b) (ht : ((fun b => Finset.image (fun a => f a b) s) '' ↑t).PairwiseDisjoint id) : s.card ∣ (Finset.image₂ f s t).card
• List.pairwiseDisjoint_consFixedLengthDigits 📋 Mathlib.Data.Nat.Digits.Lemmas
  {b : ℕ} (hb : 1 < b) (l : ℕ) : (↑(Finset.range b)).PairwiseDisjoint fun d => List.consFixedLengthDigits hb l d
• Set.PairwiseDisjoint.exists_mem_filter 📋 Mathlib.Order.Filter.Finite
  {α : Type u} {ι : Type u_1} {l : ι → Filter α} {t : Set ι} (hd : t.PairwiseDisjoint l) (ht : t.Finite) : ∃ s, (∀ (i : ι), s i ∈ l i) ∧ t.PairwiseDisjoint s
• Set.PairwiseDisjoint.exists_mem_filter_basis 📋 Mathlib.Order.Filter.Bases.Finite
  {α : Type u_1} {I : Type u_4} {l : I → Filter α} {ι : I → Sort u_5} {p : (i : I) → ι i → Prop} {s : (i : I) → ι i → Set α} {S : Set I} (hd : S.PairwiseDisjoint l) (hS : S.Finite) (h : ∀ (i : I), (l i).HasBasis (p i) (s i)) : ∃ ind, (∀ (i : I), p i (ind i)) ∧ S.PairwiseDisjoint fun i => s i (ind i)
• Finset.indicator_biUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Indicator
  {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β} (hs : (↑s).PairwiseDisjoint t) : (⋃ i ∈ s, t i).indicator f = fun a => ∑ i ∈ s, (t i).indicator f a
• Finset.indicator_biUnion_apply 📋 Mathlib.Algebra.BigOperators.Group.Finset.Indicator
  {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β} (h : (↑s).PairwiseDisjoint t) (x : κ) : (⋃ i ∈ s, t i).indicator f x = ∑ i ∈ s, (t i).indicator f x
• Finset.mulIndicator_biUnion 📋 Mathlib.Algebra.BigOperators.Group.Finset.Indicator
  {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β} (hs : (↑s).PairwiseDisjoint t) : (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a
• Finset.mulIndicator_biUnion_apply 📋 Mathlib.Algebra.BigOperators.Group.Finset.Indicator
  {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β} (h : (↑s).PairwiseDisjoint t) (x : κ) : (⋃ i ∈ s, t i).mulIndicator f x = ∏ i ∈ s, (t i).mulIndicator f x
• Finset.dens_disjiUnion 📋 Mathlib.Algebra.BigOperators.Field
  {α : Type u_3} {β : Type u_4} [Fintype β] (s : Finset α) (t : α → Finset β) (h : (↑s).PairwiseDisjoint t) : (s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens
• Finset.dens_biUnion 📋 Mathlib.Algebra.BigOperators.Field
  {α : Type u_3} {β : Type u_4} [Fintype β] {s : Finset α} {t : α → Finset β} [DecidableEq β] (h : (↑s).PairwiseDisjoint t) : (s.biUnion t).dens = ∑ u ∈ s, (t u).dens
• Finset.intervalGapsWithin_pairwiseDisjoint_Ioc 📋 Mathlib.Order.Interval.Finset.Gaps
  {α : Type u_1} [LinearOrder α] (F : Finset (α × α)) {k : ℕ} (h : F.card = k) {a b : α} (hFab : ∀ ⦃z : α × α⦄, z ∈ F → a ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b) : (Set.Iio (k + 1)).PairwiseDisjoint fun j => Set.Ioc (F.intervalGapsWithin h a b ↑j).1 (F.intervalGapsWithin h a b ↑j).2
• Finset.intervalGapsWithin_fst_le_snd 📋 Mathlib.Order.Interval.Finset.Gaps
  {α : Type u_1} [LinearOrder α] (F : Finset (α × α)) {k : ℕ} (h : F.card = k) (j : ℕ) {a b : α} (hab : a ≤ b) (hFab : ∀ ⦃z : α × α⦄, z ∈ F → a ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b) (hF : (↑F).PairwiseDisjoint fun z => Set.Icc z.1 z.2) : (F.intervalGapsWithin h a b ↑j).1 ≤ (F.intervalGapsWithin h a b ↑j).2
• Finset.pairwiseDisjoint_piAntidiag_map_addRightEmbedding 📋 Mathlib.Algebra.Order.Antidiag.Pi
  {ι : Type u_1} {μ : Type u_2} [DecidableEq ι] [AddCancelCommMonoid μ] [Finset.HasAntidiagonal μ] [DecidableEq μ] {i : ι} {s : Finset ι} (hi : i ∉ s) (n : μ) : (↑(Finset.HasAntidiagonal.antidiagonal n)).PairwiseDisjoint fun p => Finset.map (addRightEmbedding fun j => if j = i then p.1 else 0) (s.piAntidiag p.2)
• Finset.product_self_eq_disjiUnion_perm_aux 📋 Mathlib.GroupTheory.Perm.Cycle.Basic
  {α : Type u_2} {f : Equiv.Perm α} {s : Finset α} (hf : f.IsCycleOn ↑s) : (↑(Finset.range s.card)).PairwiseDisjoint fun k => Finset.map { toFun := fun i => (i, (f ^ k) i), inj' := ⋯ } s
• Set.PairwiseDisjoint.countable_of_nonempty_interior 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, (interior (s i)).Nonempty) : a.Countable
• Set.PairwiseDisjoint.countable_of_isOpen 📋 Mathlib.Topology.Bases
  {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ho : ∀ i ∈ a, IsOpen (s i)) (hne : ∀ i ∈ a, (s i).Nonempty) : a.Countable
• Set.pairwiseDisjoint_nhds 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] [T2Space X] (s : Set X) : s.PairwiseDisjoint nhds
• Set.Finite.t2_separation 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} [TopologicalSpace X] [T2Space X] {s : Set X} (hs : s.Finite) : ∃ U, (∀ (x : X), x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U
• Set.PairwiseDisjoint.countable_of_Ioo 📋 Mathlib.Topology.Order.Basic
  {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {y : α → α} {s : Set α} (h : s.PairwiseDisjoint fun x => Set.Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable
• Finset.pairwiseDisjoint_smul_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} {β : Type u_3} [SMul α β] [IsLeftCancelSMul α β] [DecidableEq β] {s : Set α} {t : Finset β} : (s.PairwiseDisjoint fun x => x • t) ↔ Set.InjOn (fun p => p.1 • p.2) (s ×ˢ ↑t)
• Finset.pairwiseDisjoint_vadd_iff 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} {β : Type u_3} [VAdd α β] [IsLeftCancelVAdd α β] [DecidableEq β] {s : Set α} {t : Finset β} : (s.PairwiseDisjoint fun x => x +ᵥ t) ↔ Set.InjOn (fun p => p.1 +ᵥ p.2) (s ×ˢ ↑t)
• Finset.card_dvd_card_add_left 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α} : ((fun b => Finset.image (fun a => a + b) s) '' ↑t).PairwiseDisjoint id → s.card ∣ (s + t).card
• Finset.card_dvd_card_mul_left 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α} : ((fun b => Finset.image (fun a => a * b) s) '' ↑t).PairwiseDisjoint id → s.card ∣ (s * t).card
• Finset.card_dvd_card_add_right 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α} : ((fun x => x +ᵥ t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s + t).card
• Finset.card_dvd_card_mul_right 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α} : ((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s * t).card
• Finset.card_dvd_card_smul_right 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {t : Finset β} {s : Finset α} : ((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s • t).card
• Finset.card_dvd_card_vadd_right 📋 Mathlib.Algebra.Group.Action.Pointwise.Finset
  {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {t : Finset β} {s : Finset α} : ((fun x => x +ᵥ t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s +ᵥ t).card
• Finset.card_dvd_card_smul_right₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset
  {α : Type u_1} {β : Type u_2} [DecidableEq β] [GroupWithZero α] [MulAction α β] {t : Finset β} {s : Finset α} (hs : ∀ a ∈ s, a ≠ 0) : ((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s • t).card
• Finset.pairwiseDisjoint_smul_iff₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset
  {α : Type u_1} {β : Type u_2} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s : Set α} {t : Finset β} (hs : ∀ a ∈ s, a ≠ 0) : (s.PairwiseDisjoint fun x => x • t) ↔ Set.InjOn (fun p => p.1 • p.2) (s ×ˢ ↑t)
• ENNReal.tsum_biUnion 📋 Mathlib.Topology.Algebra.InfiniteSum.ENNReal
  {α : Type u_1} {ι : Type u_4} {f : α → ENNReal} {t : ι → Set α} (h : Set.univ.PairwiseDisjoint t) : ∑' (x : ↑(⋃ i, t i)), f ↑x = ∑' (i : ι) (x : ↑(t i)), f ↑x
• ENNReal.tsum_biUnion' 📋 Mathlib.Topology.Algebra.InfiniteSum.ENNReal
  {α : Type u_1} {ι : Type u_4} {S : Set ι} {f : α → ENNReal} {t : ι → Set α} (h : S.PairwiseDisjoint t) : ∑' (x : ↑(⋃ i ∈ S, t i)), f ↑x = ∑' (i : ↑S) (x : ↑(t ↑i)), f ↑x
• Set.PairwiseDisjoint.aedisjoint 📋 Mathlib.MeasureTheory.Measure.AEDisjoint
  {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} {s : Set ι} (hf : s.PairwiseDisjoint f) : s.Pairwise (Function.onFun (MeasureTheory.AEDisjoint μ) f)
• MeasureTheory.measure_biUnion_finset 📋 Mathlib.MeasureTheory.Measure.Basic
  {α : Type u_1} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).PairwiseDisjoint f) (hm : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p)
• MeasureTheory.measure_biUnion 📋 Mathlib.MeasureTheory.Measure.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.PairwiseDisjoint f) (h : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)
• MeasureTheory.lintegral_biUnion_finset 📋 Mathlib.MeasureTheory.Integral.Lebesgue.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α} (hd : (↑s).PairwiseDisjoint t) (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ (a : α) in t b, f a ∂μ
• MeasureTheory.lintegral_biUnion 📋 Mathlib.MeasureTheory.Integral.Lebesgue.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ENNReal) : ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ
• AddSubgroup.IsComplement.pairwiseDisjoint_vadd 📋 Mathlib.GroupTheory.Complement
  {G : Type u_1} [AddGroup G] {S T : Set G} (hst : AddSubgroup.IsComplement S T) : S.PairwiseDisjoint fun x => x +ᵥ T
• Subgroup.IsComplement.pairwiseDisjoint_smul 📋 Mathlib.GroupTheory.Complement
  {G : Type u_1} [Group G] {S T : Set G} (hst : Subgroup.IsComplement S T) : S.PairwiseDisjoint fun x => x • T
• MeasureTheory.sum_measureReal_le_measureReal_univ 📋 Mathlib.MeasureTheory.Measure.Real
  {α : Type u_1} {ι : Type u_3} {x✝ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasurableSet (t i)) (H : (↑s).PairwiseDisjoint t) : ∑ i ∈ s, μ.real (t i) ≤ μ.real Set.univ
• MeasureTheory.measureReal_biUnion_finset 📋 Mathlib.MeasureTheory.Measure.Real
  {α : Type u_1} {ι : Type u_3} {x✝ : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).PairwiseDisjoint f) (hm : ∀ b ∈ s, MeasurableSet (f b)) (h : ∀ b ∈ s, μ (f b) ≠ ⊤ := by finiteness) : μ.real (⋃ b ∈ s, f b) = ∑ p ∈ s, μ.real (f p)
• OrderedFinpartition.disjoint 📋 Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
  {n : ℕ} (self : OrderedFinpartition n) : Set.univ.PairwiseDisjoint fun m => Set.range (self.emb m)
• OrderedFinpartition.mk 📋 Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
  {n : ℕ} (length : ℕ) (partSize : Fin length → ℕ) (partSize_pos : ∀ (m : Fin length), 0 < partSize m) (emb : (m : Fin length) → Fin (partSize m) → Fin n) (emb_strictMono : ∀ (m : Fin length), StrictMono (emb m)) (parts_strictMono : StrictMono fun m => emb m ⟨partSize m - 1, ⋯⟩) (disjoint : Set.univ.PairwiseDisjoint fun m => Set.range (emb m)) (cover : ∀ (x : Fin n), ∃ m, x ∈ Set.range (emb m)) : OrderedFinpartition n
• VitaliFamily.FineSubfamilyOn.covering_disjoint 📋 Mathlib.MeasureTheory.Covering.VitaliFamily
  {X : Type u_1} [PseudoMetricSpace X] {m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} {v : VitaliFamily μ} {f : X → Set (Set X)} {s : Set X} (h : v.FineSubfamilyOn f s) : h.index.PairwiseDisjoint h.covering
• VitaliFamily.FineSubfamilyOn.exists_disjoint_covering_ae 📋 Mathlib.MeasureTheory.Covering.VitaliFamily
  {X : Type u_1} [PseudoMetricSpace X] {m0 : MeasurableSpace X} {μ : MeasureTheory.Measure X} {v : VitaliFamily μ} {f : X → Set (Set X)} {s : Set X} (h : v.FineSubfamilyOn f s) : ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ v.setsAt p.1 ∩ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
• VitaliFamily.covering 📋 Mathlib.MeasureTheory.Covering.VitaliFamily
  {X : Type u_1} [PseudoMetricSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} (self : VitaliFamily μ) (s : Set X) (f : X → Set (Set X)) : (∀ x ∈ s, f x ⊆ self.setsAt x) → (∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ Metric.closedBall x ε) → ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
• VitaliFamily.mk 📋 Mathlib.MeasureTheory.Covering.VitaliFamily
  {X : Type u_1} [PseudoMetricSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} (setsAt : X → Set (Set X)) (measurableSet : ∀ (x : X), ∀ s ∈ setsAt x, MeasurableSet s) (nonempty_interior : ∀ (x : X), ∀ s ∈ setsAt x, (interior s).Nonempty) (nontrivial : ∀ (x : X), ∀ ε > 0, ∃ s ∈ setsAt x, s ⊆ Metric.closedBall x ε) (covering : ∀ (s : Set X) (f : X → Set (Set X)), (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ Metric.closedBall x ε) → ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0) : VitaliFamily μ
• Vitali.exists_disjoint_subfamily_covering_enlargement_ball 📋 Mathlib.MeasureTheory.Covering.Vitali
  {α : Type u_1} {ι : Type u_2} [PseudoMetricSpace α] (t : Set ι) (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) (τ : ℝ) (hτ : 3 < τ) : ∃ u ⊆ t, (u.PairwiseDisjoint fun a => Metric.ball (x a) (r a)) ∧ ∀ a ∈ t, ∃ b ∈ u, Metric.ball (x a) (r a) ⊆ Metric.ball (x b) (τ * r b)
• Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall 📋 Mathlib.MeasureTheory.Covering.Vitali
  {α : Type u_1} {ι : Type u_2} [PseudoMetricSpace α] (t : Set ι) (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) (τ : ℝ) (hτ : 3 < τ) : ∃ u ⊆ t, (u.PairwiseDisjoint fun a => Metric.closedBall (x a) (r a)) ∧ ∀ a ∈ t, ∃ b ∈ u, Metric.closedBall (x a) (r a) ⊆ Metric.closedBall (x b) (τ * r b)
• Vitali.exists_disjoint_subfamily_covering_enlargement 📋 Mathlib.MeasureTheory.Covering.Vitali
  {α : Type u_1} {ι : Type u_2} (B : ι → Set α) (t : Set ι) (δ : ι → ℝ) (τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R) (hne : ∀ a ∈ t, (B a).Nonempty) : ∃ u ⊆ t, u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b
• Vitali.exists_disjoint_covering_ae 📋 Mathlib.MeasureTheory.Covering.Vitali
  {α : Type u_1} {ι : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι) (C : NNReal) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ Metric.closedBall (c a) (r a)) (μB : ∀ a ∈ t, μ (Metric.closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)) (ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a)) (hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x) : ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
• Vitali.exists_disjoint_covering_ae' 📋 Mathlib.MeasureTheory.Covering.Vitali
  {α : Type u_1} {ι : Type u_2} [PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι) (C : NNReal) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ Metric.closedBall (c a) (r a)) (μB : ∀ a ∈ t, μ (Metric.closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)) (ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a)) (hf : ∀ x ∈ s, ∃ᶠ (ε : ℝ) in nhdsWithin 0 (Set.Ioi 0), ∃ a ∈ t, r a = ε ∧ c a = x) : ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
• Besicovitch.exist_disjoint_covering_families 📋 Mathlib.MeasureTheory.Covering.Besicovitch
  {α : Type u_1} [MetricSpace α] {β : Type u} {N : ℕ} {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (Besicovitch.SatelliteConfig α N τ)) (q : Besicovitch.BallPackage β α) : ∃ s, (∀ (i : Fin N), (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧ Set.range q.c ⊆ ⋃ i, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)
• Besicovitch.exists_disjoint_closedBall_covering_ae 📋 Mathlib.MeasureTheory.Covering.Besicovitch
  {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty) (R : α → ℝ) (hR : ∀ x ∈ s, 0 < R x) : ∃ t r, t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x ∩ Set.Ioo 0 (R x)) ∧ μ (s \ ⋃ x ∈ t, Metric.closedBall x (r x)) = 0 ∧ t.PairwiseDisjoint fun x => Metric.closedBall x (r x)
• Besicovitch.exists_disjoint_closedBall_covering_ae_aux 📋 Mathlib.MeasureTheory.Covering.Besicovitch
  {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty) : ∃ t, t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, Metric.closedBall p.1 p.2) = 0 ∧ t.PairwiseDisjoint fun p => Metric.closedBall p.1 p.2
• Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux 📋 Mathlib.MeasureTheory.Covering.Besicovitch
  {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty) : ∃ t, t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, Metric.closedBall p.1 p.2) = 0 ∧ t.PairwiseDisjoint fun p => Metric.closedBall p.1 p.2
• Besicovitch.exist_finset_disjoint_balls_large_measure 📋 Mathlib.MeasureTheory.Covering.Besicovitch
  {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {N : ℕ} {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (Besicovitch.SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ) (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) : ∃ t, ↑t ⊆ s ∧ μ (s \ ⋃ x ∈ t, Metric.closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s ∧ (↑t).PairwiseDisjoint fun x => Metric.closedBall x (r x)
• AbsolutelyContinuousOnInterval.dist_le_of_pairwiseDisjoint_hasSum 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
  {X : Type u_1} [PseudoMetricSpace X] {f : ℝ → X} {d b y : ℝ} (hdb : d ≤ b) (hf : AbsolutelyContinuousOnInterval f d b) {u : Set (ℝ × ℝ)} (hu₁ : ∀ z ∈ u, d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) (hu₂ : u.PairwiseDisjoint fun z => Set.Icc z.1 z.2) (hu₃ : HasSum (fun z => (↑z).2 - (↑z).1) (b - d)) (hu₄ : HasSum (fun z => dist (f (↑z).1) (f (↑z).2)) y) : dist (f d) (f b) ≤ y
• exists_dist_slope_lt_pairwiseDisjoint_hasSum 📋 Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
  {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f f' : ℝ → F} {d b η : ℝ} (hdb : d ≤ b) (hf : ∀ᵐ (x : ℝ), x ∈ Set.Ioo d b → HasDerivAt f (f' x) x) (hη : 0 < η) : ∃ u, (∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η) ∧ (u.PairwiseDisjoint fun z => Set.Icc z.1 z.2) ∧ HasSum (fun z => (↑z).2 - (↑z).1) (b - d)
• Finset.pairwiseDisjoint_range_singleton 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_1} : (Set.range singleton).PairwiseDisjoint id
• List.pairwise_disjoint_of_coe_toFinset_pairwiseDisjoint 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_4} {ι : Type u_5} [PartialOrder α] [OrderBot α] [DecidableEq ι] {l : List ι} {f : ι → α} (hl : (↑l.toFinset).PairwiseDisjoint f) (hn : l.Nodup) : List.Pairwise (Function.onFun Disjoint f) l
• List.pairwiseDisjoint_iff_coe_toFinset_pairwise_disjoint 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_4} {ι : Type u_5} [PartialOrder α] [OrderBot α] [DecidableEq ι] {l : List ι} {f : ι → α} (hn : l.Nodup) : (↑l.toFinset).PairwiseDisjoint f ↔ List.Pairwise (Function.onFun Disjoint f) l
• Set.PairwiseDisjoint.elim_finset 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_1} {ι : Type u_2} {s : Set ι} {f : ι → Finset α} (hs : s.PairwiseDisjoint f) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (a : α) (hai : a ∈ f i) (haj : a ∈ f j) : i = j
• Set.PairwiseDisjoint.image_finset_of_le 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_1} {ι : Type u_2} [SemilatticeInf α] [OrderBot α] [DecidableEq ι] {s : Finset ι} {f : ι → α} (hs : (↑s).PairwiseDisjoint f) {g : ι → ι} (hf : ∀ (a : ι), f (g a) ≤ f a) : (↑(Finset.image g s)).PairwiseDisjoint f
• Set.PairwiseDisjoint.biUnion_finset 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [Lattice α] [OrderBot α] {s : Set ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.PairwiseDisjoint fun i' => (g i').sup f) (hg : ∀ i ∈ s, (↑(g i)).PairwiseDisjoint f) : (⋃ i ∈ s, ↑(g i)).PairwiseDisjoint f
• Set.PairwiseDisjoint.attach 📋 Mathlib.Data.Finset.Pairwise
  {α : Type u_1} {ι : Type u_2} [SemilatticeInf α] [OrderBot α] {s : Finset ι} {f : ι → α} (hs : (↑s).PairwiseDisjoint f) : (↑s.attach).PairwiseDisjoint (f ∘ Subtype.val)
• Finpartition.disjoint 📋 Mathlib.Order.Partition.Finpartition
  {α : Type u_1} [Lattice α] [OrderBot α] {a : α} (P : Finpartition a) : (↑P.parts).PairwiseDisjoint id
• Finpartition.ofPairwiseDisjoint 📋 Mathlib.Order.Partition.Finpartition
  {α : Type u_1} [DistribLattice α] [OrderBot α] [DecidableEq α] (parts : Finset α) (hdisjoint : (↑parts).PairwiseDisjoint id) : Finpartition (parts.sup id)
• Finpartition.ofPairwiseDisjoint_parts 📋 Mathlib.Order.Partition.Finpartition
  {α : Type u_1} [DistribLattice α] [OrderBot α] [DecidableEq α] (parts : Finset α) (hdisjoint : (↑parts).PairwiseDisjoint id) : (Finpartition.ofPairwiseDisjoint parts hdisjoint).parts = parts.erase ⊥
• Finpartition.pairwiseDisjoint_apply 📋 Mathlib.Order.Partition.Finpartition
  {α : Type u_1} [Lattice α] [OrderBot α] {a : α} (P : Finpartition a) {β : Type u_2} {f : α → β} [SemilatticeInf β] [OrderBot β] (hf : ∀ (x y : α), f (x ⊓ y) = f x ⊓ f y) (hbot : f ⊥ = ⊥) : (↑P.parts).PairwiseDisjoint f
• Finpartition.sum_ofPairwiseDisjoint_eq_sum 📋 Mathlib.Order.Partition.Finpartition
  {α : Type u_1} [DistribLattice α] [OrderBot α] [DecidableEq α] {parts : Finset α} (hdisjoint : (↑parts).PairwiseDisjoint id) {X : Type u_2} [AddCommMonoid X] {f : α → X} (hf : f ⊥ = 0) : ∑ p ∈ (Finpartition.ofPairwiseDisjoint parts hdisjoint).parts, f p = ∑ p ∈ parts, f p
• Finset.pairwiseDisjoint_slice 📋 Mathlib.Data.Finset.Slice
  {α : Type u_1} {𝒜 : Finset (Finset α)} : Set.univ.PairwiseDisjoint 𝒜.slice
• Setoid.IsPartition.pairwiseDisjoint 📋 Mathlib.Data.Setoid.Partition
  {α : Type u_1} {c : Set (Set α)} (hc : Setoid.IsPartition c) : c.PairwiseDisjoint id
• Setoid.setoidOfDisjointUnion 📋 Mathlib.Data.Setoid.Partition
  {α : Type u_1} {c : Set (Set α)} (hu : ⋃₀ c = Set.univ) (H : c.PairwiseDisjoint id) : Setoid α
• Setoid.eqv_classes_disjoint 📋 Mathlib.Data.Setoid.Partition
  {α : Type u_1} {c : Set (Set α)} (H : ∀ (a : α), ∃! b, b ∈ c ∧ a ∈ b) : c.PairwiseDisjoint id
• IndexedPartition.piecewise_inj 📋 Mathlib.Data.Setoid.Partition
  {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (hs : IndexedPartition s) {β : Type u_3} {f : ι → α → β} (h_injOn : ∀ (i : ι), Set.InjOn (f i) (s i)) (h_disjoint : Set.univ.PairwiseDisjoint fun i => f i '' s i) : Function.Injective (hs.piecewise f)
• Setoid.eqv_classes_of_disjoint_union 📋 Mathlib.Data.Setoid.Partition
  {α : Type u_1} {c : Set (Set α)} (hu : ⋃₀ c = Set.univ) (H : c.PairwiseDisjoint id) (a : α) : ∃! b, b ∈ c ∧ a ∈ b
• Set.PairwiseDisjoint.isPartition_of_exists_of_ne_empty 📋 Mathlib.Data.Setoid.Partition
  {α : Type u_2} {s : Set (Set α)} (h₁ : s.PairwiseDisjoint id) (h₂ : ∀ (a : α), ∃ x ∈ s, a ∈ x) (h₃ : ∅ ∉ s) : Setoid.IsPartition s
• SimpleGraph.EdgeLabeling.pairwiseDisjoint_univ_labelGraph 📋 Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling
  {V : Type u_1} {G : SimpleGraph V} {K : Type u_3} {C : G.EdgeLabeling K} : Set.univ.PairwiseDisjoint C.labelGraph
• Set.Finite.encard_biUnion 📋 Mathlib.Data.Set.Card.Arithmetic
  {α : Type u_1} {ι : Type u_2} {t : Set ι} (ht : t.Finite) {s : ι → Set α} (hs : t.PairwiseDisjoint s) : (⋃ i ∈ t, s i).encard = ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard
• Set.Finite.ncard_biUnion 📋 Mathlib.Data.Set.Card.Arithmetic
  {α : Type u_1} {ι : Type u_2} {t : Set ι} (ht : t.Finite) {s : ι → Set α} (hs : ∀ i ∈ t, (s i).Finite) (h : t.PairwiseDisjoint s) : (⋃ i ∈ t, s i).ncard = ∑ᶠ (i : ι) (_ : i ∈ t), (s i).ncard
• exists_clopen_partition_of_clopen_cover 📋 Mathlib.Topology.Separation.Profinite
  {X : Type u_3} {I : Type u_4} [TopologicalSpace X] [CompactSpace X] [T2Space X] [TotallyDisconnectedSpace X] [Finite I] {Z D : I → Set X} (Z_closed : ∀ (i : I), IsClosed (Z i)) (D_clopen : ∀ (i : I), IsClopen (D i)) (Z_subset_D : ∀ (i : I), Z i ⊆ D i) (Z_disj : Set.univ.PairwiseDisjoint Z) : ∃ C, (∀ (i : I), IsClopen (C i)) ∧ (∀ (i : I), Z i ⊆ C i) ∧ (∀ (i : I), C i ⊆ D i) ∧ ⋃ i, D i ⊆ ⋃ i, C i ∧ Set.univ.PairwiseDisjoint C
• Finset.support_sum_eq 📋 Mathlib.Data.Finsupp.BigOperators
  {ι : Type u_1} {M : Type u_2} [DecidableEq ι] [AddCommMonoid M] (s : Finset (ι →₀ M)) (hs : (↑s).PairwiseDisjoint Finsupp.support) : (s.sum id).support = s.sup Finsupp.support
• IsChain.pairwiseDisjoint_sUnion 📋 Mathlib.Data.Set.Pairwise.Chain
  {α : Type u_1} {β : Type u_2} {c : Set (Set α)} (hc : IsChain (fun x1 x2 => x1 ⊆ x2) c) [PartialOrder β] [OrderBot β] (f : α → β) : (⋃₀ c).PairwiseDisjoint f ↔ ∀ s ∈ c, s.PairwiseDisjoint f
• IsChain.pairwiseDisjoint_iUnion₂ 📋 Mathlib.Data.Set.Pairwise.Chain
  {α : Type u_1} {β : Type u_2} {c : Set (Set α)} (hc : IsChain (fun x1 x2 => x1 ⊆ x2) c) [PartialOrder β] [OrderBot β] (f : α → β) : (⋃ s ∈ c, s).PairwiseDisjoint f ↔ ∀ s ∈ c, s.PairwiseDisjoint f
• AddSubgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {s : Finset ι} {H : AddSubgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑H = Set.univ) (hind : H.index = s.card) : (↑s).PairwiseDisjoint fun x => g x +ᵥ ↑H
• Subgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [Group G] {ι : Type u_2} {s : Finset ι} {H : Subgroup G} {g : ι → G} (hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ) (hind : H.index = s.card) : (↑s).PairwiseDisjoint fun x => g x • ↑H
• AddSubgroup.pairwiseDisjoint_leftCoset_cover_of_sum_neg_index_eq_zero 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidablePred AddSubgroup.FiniteIndex] : ∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑({i ∈ s | (H i).FiniteIndex})).PairwiseDisjoint fun i => g i +ᵥ ↑(H i)
• Subgroup.pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidablePred Subgroup.FiniteIndex] : ∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑({i ∈ s | (H i).FiniteIndex})).PairwiseDisjoint fun i => g i • ↑(H i)
• AddSubgroup.leftCoset_cover_filter_FiniteIndex_aux 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ) [DecidablePred AddSubgroup.FiniteIndex] : ⋃ k ∈ {i ∈ s | (H i).FiniteIndex}, g k +ᵥ ↑(H k) = Set.univ ∧ 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹ ∧ (∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑({i ∈ s | (H i).FiniteIndex})).PairwiseDisjoint fun i => g i +ᵥ ↑(H i))
• Subgroup.leftCoset_cover_filter_FiniteIndex_aux 📋 Mathlib.GroupTheory.CosetCover
  {G : Type u_1} [Group G] {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι} (hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ) [DecidablePred Subgroup.FiniteIndex] : ⋃ k ∈ {i ∈ s | (H i).FiniteIndex}, g k • ↑(H k) = Set.univ ∧ 1 ≤ ∑ i ∈ s, (↑(H i).index)⁻¹ ∧ (∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑({i ∈ s | (H i).FiniteIndex})).PairwiseDisjoint fun i => g i • ↑(H i))
• AddAction.IsBlock.pairwiseDisjoint_range_vadd 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} {X : Type u_2} [VAdd G X] {B : Set X} : AddAction.IsBlock G B → (Set.range fun g => g +ᵥ B).PairwiseDisjoint id
• MulAction.IsBlock.pairwiseDisjoint_range_smul 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X} : MulAction.IsBlock G B → (Set.range fun g => g • B).PairwiseDisjoint id
• AddAction.isBlock_iff_pairwiseDisjoint_range_vadd 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} {X : Type u_2} [VAdd G X] {B : Set X} : AddAction.IsBlock G B ↔ (Set.range fun g => g +ᵥ B).PairwiseDisjoint id
• MulAction.isBlock_iff_pairwiseDisjoint_range_smul 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} {X : Type u_2} [SMul G X] {B : Set X} : MulAction.IsBlock G B ↔ (Set.range fun g => g • B).PairwiseDisjoint id
• AddAction.orbit.pairwiseDisjoint 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} [AddGroup G] {X : Type u_2} [AddAction G X] : (Set.range fun x => AddAction.orbit G x).PairwiseDisjoint id
• MulAction.orbit.pairwiseDisjoint 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_1} [Group G] {X : Type u_2} [MulAction G X] : (Set.range fun x => MulAction.orbit G x).PairwiseDisjoint id
• MeasureTheory.VectorMeasure.of_biUnion_finset 📋 Mathlib.MeasureTheory.VectorMeasure.Basic
  {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {v : MeasureTheory.VectorMeasure α M} [T2Space M] {ι : Type u_4} {s : Finset ι} {f : ι → Set α} (hd : (↑s).PairwiseDisjoint f) (hm : ∀ b ∈ s, MeasurableSet (f b)) : v (⋃ b ∈ s, f b) = ∑ p ∈ s, v (f p)
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfUnion 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {J : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hJ : ↑J ⊆ C) : (↑J).PairwiseDisjoint (hC.disjointOfUnion hJ)
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfDiff 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) : (↑(hC.disjointOfDiff hs ht)).PairwiseDisjoint id
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfDiffUnion 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) : (↑(hC.disjointOfDiffUnion hs hI)).PairwiseDisjoint id
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_insert_disjointOfDiff 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) : (insert t ↑(hC.disjointOfDiff hs ht)).PairwiseDisjoint id
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_disjointOfUnion_of_mem 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {j : Set α} {J : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) : (↑(hC.disjointOfUnion hJ j)).PairwiseDisjoint id
• MeasureTheory.IsSetSemiring.pairwiseDisjoint_biUnion_disjointOfUnion 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {J : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hJ : ↑J ⊆ C) : (⋃ x ∈ J, ↑(hC.disjointOfUnion hJ x)).PairwiseDisjoint id
• MeasureTheory.IsSetSemiring.sdiff_eq_sUnion' 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetSemiring C) (s : Set α) : s ∈ C → ∀ t ∈ C, ∃ I, ↑I ⊆ C ∧ (↑I).PairwiseDisjoint id ∧ s \ t = ⋃₀ ↑I
• MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq 📋 Mathlib.MeasureTheory.SetSemiring
  {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) : ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I = ⋃₀ ↑J

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59