Loogle!

Result

Found 342 declarations mentioning Set.sUnion. Of these, only the first 200 are shown.

Set.sUnion 📋 Mathlib.Order.SetNotation
{α : Type u} (S : Set (Set α)) : Set α
Set.sSup_eq_sUnion 📋 Mathlib.Order.SetNotation
{α : Type u} (S : Set (Set α)) : sSup S = ⋃₀ S
Set.mem_sUnion 📋 Mathlib.Order.SetNotation
{α : Type u} {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t
Set.sUnion_symmDiff_subset 📋 Mathlib.Data.Set.Lattice.Indexed
{α : Type u_1} {s : Set α} {S : Set (Set α)} (hS : S.Nonempty) : symmDiff (⋃₀ S) s ⊆ ⋃₀ ((fun x => symmDiff x s) '' S)
Set.symmDiff_sUnion_subset 📋 Mathlib.Data.Set.Lattice.Indexed
{α : Type u_1} {s : Set α} {S : Set (Set α)} (hS : S.Nonempty) : symmDiff s (⋃₀ S) ⊆ ⋃₀ ((fun x => symmDiff s x) '' S)
Set.sUnion_symmDiff_sUnion_subset 📋 Mathlib.Data.Set.Lattice.Indexed
{α : Type u_1} {S T : Set (Set α)} (hS : S.Nonempty) (hT : T.Nonempty) : symmDiff (⋃₀ S) (⋃₀ T) ⊆ ⋃₀ Set.image2 (fun x1 x2 => symmDiff x1 x2) S T
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)
Set.Nonempty.of_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} (h : (⋃₀ s).Nonempty) : s.Nonempty
Set.sUnion_empty 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} : ⋃₀ ∅ = ∅
Set.Nonempty.of_sUnion_eq_univ 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} [Nonempty α] {s : Set (Set α)} (h : ⋃₀ s = Set.univ) : s.Nonempty
Set.sUnion_singleton 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (s : Set α) : ⋃₀ {s} = s
Set.sUnion_range 📋 Mathlib.Data.Set.Lattice.Bounded
{β : Type u_2} {ι : Sort u_4} (f : ι → Set β) : ⋃₀ Set.range f = ⋃ x, f x
Set.sUnion_iUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {ι : Sort u_4} (s : ι → Set (Set α)) : ⋃₀ ⋃ i, s i = ⋃ i, ⋃₀ s i
Set.subset_sUnion_of_mem 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀ S
Set.sUnion_mono 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀ S ⊆ ⋃₀ T
Set.sUnion_subset_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀ S ⊆ ⋃₀ T
Set.subset_powerset_iff 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t
Set.compl_sInter 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀ (compl '' S)
Set.compl_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (S : Set (Set α)) : (⋃₀ S)ᶜ = ⋂₀ (compl '' S)
Set.sInter_eq_compl_sUnion_compl 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (S : Set (Set α)) : ⋂₀ S = (⋃₀ (compl '' S))ᶜ
Set.sUnion_eq_compl_sInter_compl 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (S : Set (Set α)) : ⋃₀ S = (⋂₀ (compl '' S))ᶜ
Set.nonempty_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S : Set (Set α)} : (⋃₀ S).Nonempty ↔ ∃ s ∈ S, s.Nonempty
Set.sUnion_insert 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (s : Set α) (T : Set (Set α)) : ⋃₀ insert s T = s ∪ ⋃₀ T
Set.sUnion_union 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (S T : Set (Set α)) : ⋃₀ (S ∪ T) = ⋃₀ S ∪ ⋃₀ T
Set.sUnion_diff_singleton_empty 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (s : Set (Set α)) : ⋃₀ (s \ {∅}) = ⋃₀ s
Set.sUnion_sdiff_singleton_empty 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (s : Set (Set α)) : ⋃₀ (s \ {∅}) = ⋃₀ s
Set.sUnion_mono_subsets 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ (t : Set α), t ⊆ f t) : ⋃₀ s ⊆ ⋃₀ (f '' s)
Set.sUnion_mono_supsets 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ (t : Set α), f t ⊆ t) : ⋃₀ (f '' s) ⊆ ⋃₀ s
Set.sUnion_subset 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀ S ⊆ t
Set.subset_sUnion_of_subset 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u) (h₂ : u ∈ t) : s ⊆ ⋃₀ t
Set.mem_sUnion_of_mem 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) : x ∈ ⋃₀ S
Set.sUnion_subset_iff 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} {t : Set α} : ⋃₀ s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t
Set.sUnion_eq_iUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} : ⋃₀ s = ⋃ i, ↑i
Set.sUnion_pair 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} (s t : Set α) : ⋃₀ {s, t} = s ∪ t
Set.notMem_of_notMem_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀ S) (ht : t ∈ S) : x ∉ t
Set.sUnion_eq_empty 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S : Set (Set α)} : ⋃₀ S = ∅ ↔ ∀ s ∈ S, s = ∅
Set.sUnion_eq_univ_iff 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {c : Set (Set α)} : ⋃₀ c = Set.univ ↔ ∀ (a : α), ∃ b ∈ c, a ∈ b
Set.sUnion_eq_biUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s : Set (Set α)} : ⋃₀ s = ⋃ i ∈ s, i
Set.sUnion_image 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {β : Type u_2} (f : α → Set β) (s : Set α) : ⋃₀ (f '' s) = ⋃ a ∈ s, f a
Set.directedOn_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {r : α → α → Prop} {S : Set (Set α)} (hd : DirectedOn (fun x1 x2 => x1 ⊆ x2) S) (h : ∀ x ∈ S, DirectedOn r x) : DirectedOn r (⋃₀ S)
Set.inter_empty_of_inter_sUnion_empty 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s t : Set α} {S : Set (Set α)} (hs : t ∈ S) (h : s ∩ ⋃₀ S = ∅) : s ∩ t = ∅
Set.sUnionPowersetGI 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} : GaloisInsertion (fun x => ⋃₀ x) fun x => 𝒫 x
Set.sUnion_powerset_gc 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} : GaloisConnection (fun x => ⋃₀ x) fun x => 𝒫 x
Set.sUnion_image2 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → Set γ) (s : Set α) (t : Set β) : ⋃₀ Set.image2 f s t = ⋃ a ∈ s, ⋃ b ∈ t, f a b
Set.sUnion_mem_empty_univ 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {S : Set (Set α)} (h : S ⊆ {∅, Set.univ}) : ⋃₀ S ∈ {∅, Set.univ}
Set.iUnion_range_eq_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_7} {β : Type u_8} (C : Set (Set α)) {f : (s : ↑C) → β → ↑↑s} (hf : ∀ (s : ↑C), Function.Surjective (f s)) : (⋃ y, Set.range fun s => ↑(f s y)) = ⋃₀ C
Set.sUnion_inter_sUnion 📋 Mathlib.Data.Set.Lattice.Bounded
{α : Type u_1} {s t : Set (Set α)} : ⋃₀ s ∩ ⋃₀ t = ⋃ p ∈ s ×ˢ t, p.1 ∩ p.2
Set.image_sUnion 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set α)} : f '' ⋃₀ s = ⋃₀ (Set.image f '' s)
Set.surjOn_sUnion 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, Set.SurjOn f s t) : Set.SurjOn f s (⋃₀ T)
Set.mapsTo_sUnion 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {S : Set (Set α)} {t : Set β} {f : α → β} : Set.MapsTo f (⋃₀ S) t ↔ ∀ s ∈ S, Set.MapsTo f s t
Set.preimage_sUnion 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀ s = ⋃ t ∈ s, f ⁻¹' t
Set.prod_sUnion 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀ C = ⋃₀ ((fun t => s ×ˢ t) '' C)
Set.sUnion_prod_const 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {C : Set (Set α)} {t : Set β} : ⋃₀ C ×ˢ t = ⋃₀ ((fun s => s ×ˢ t) '' C)
Set.image2_sUnion_left 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (S : Set (Set α)) (t : Set β) : Set.image2 f (⋃₀ S) t = ⋃ s ∈ S, Set.image2 f s t
Set.image2_sUnion_right 📋 Mathlib.Data.Set.Lattice.Image
{α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (s : Set α) (T : Set (Set β)) : Set.image2 f s (⋃₀ T) = ⋃ t ∈ T, Set.image2 f s t
Set.sUnion_inv 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Inv α] (S : Set (Set α)) : (⋃₀ S)⁻¹ = ⋃ s ∈ S, s⁻¹
Set.sUnion_neg 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Neg α] (S : Set (Set α)) : -⋃₀ S = ⋃ s ∈ S, -s
Set.sUnion_vsub 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [VSub α β] (S : Set (Set β)) (t : Set β) : ⋃₀ S -ᵥ t = ⋃ s ∈ S, s -ᵥ t
Set.vsub_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [VSub α β] (s : Set β) (T : Set (Set β)) : s -ᵥ ⋃₀ T = ⋃ t ∈ T, s -ᵥ t
Set.add_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Add α] (s : Set α) (T : Set (Set α)) : s + ⋃₀ T = ⋃ t ∈ T, s + t
Set.div_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Div α] (s : Set α) (T : Set (Set α)) : s / ⋃₀ T = ⋃ t ∈ T, s / t
Set.mul_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Mul α] (s : Set α) (T : Set (Set α)) : s * ⋃₀ T = ⋃ t ∈ T, s * t
Set.sUnion_add 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Add α] (S : Set (Set α)) (t : Set α) : ⋃₀ S + t = ⋃ s ∈ S, s + t
Set.sUnion_div 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Div α] (S : Set (Set α)) (t : Set α) : ⋃₀ S / t = ⋃ s ∈ S, s / t
Set.sUnion_mul 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Mul α] (S : Set (Set α)) (t : Set α) : ⋃₀ S * t = ⋃ s ∈ S, s * t
Set.sUnion_sub 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Sub α] (S : Set (Set α)) (t : Set α) : ⋃₀ S - t = ⋃ s ∈ S, s - t
Set.sub_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} [Sub α] (s : Set α) (T : Set (Set α)) : s - ⋃₀ T = ⋃ t ∈ T, s - t
Set.smul_set_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [SMul α β] (a : α) (S : Set (Set β)) : a • ⋃₀ S = ⋃ s ∈ S, a • s
Set.vadd_set_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [VAdd α β] (a : α) (S : Set (Set β)) : a +ᵥ ⋃₀ S = ⋃ s ∈ S, a +ᵥ s
Set.sUnion_smul 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [SMul α β] (S : Set (Set α)) (t : Set β) : ⋃₀ S • t = ⋃ s ∈ S, s • t
Set.sUnion_vadd 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [VAdd α β] (S : Set (Set α)) (t : Set β) : ⋃₀ S +ᵥ t = ⋃ s ∈ S, s +ᵥ t
Set.smul_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [SMul α β] (s : Set α) (T : Set (Set β)) : s • ⋃₀ T = ⋃ t ∈ T, s • t
Set.vadd_sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.Lattice
{α : Type u_1} {β : Type u_2} [VAdd α β] (s : Set α) (T : Set (Set β)) : s +ᵥ ⋃₀ T = ⋃ t ∈ T, s +ᵥ t
IsSelfInv.sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.SelfInv
{α : Type u_1} [Inv α] {S : Set (Set α)} (h : ∀ s ∈ S, IsSelfInv s) : IsSelfInv (⋃₀ S)
IsSelfNeg.sUnion 📋 Mathlib.Algebra.Group.Pointwise.Set.SelfInv
{α : Type u_1} [Neg α] {S : Set (Set α)} (h : ∀ s ∈ S, IsSelfNeg s) : IsSelfNeg (⋃₀ S)
Finset.sup_id_set_eq_sUnion 📋 Mathlib.Data.Finset.Lattice.Fold
{α : Type u_2} (s : Finset (Set α)) : s.sup id = ⋃₀ ↑s
Set.disjoint_sUnion_left 📋 Mathlib.Data.Set.Lattice.Disjoint
{α : Type u_1} {S : Set (Set α)} {t : Set α} : Disjoint (⋃₀ S) t ↔ ∀ s ∈ S, Disjoint s t
Set.disjoint_sUnion_right 📋 Mathlib.Data.Set.Lattice.Disjoint
{α : Type u_1} {s : Set α} {S : Set (Set α)} : Disjoint s (⋃₀ S) ↔ ∀ t ∈ S, Disjoint s t
Set.pairwise_sUnion 📋 Mathlib.Data.Set.Pairwise.Lattice
{α : Type u_1} {r : α → α → Prop} {s : Set (Set α)} (hd : DirectedOn (fun x1 x2 => x1 ⊆ x2) s) : (⋃₀ s).Pairwise r ↔ ∀ a ∈ s, a.Pairwise r
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.Infinite.sUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {s : Set (Set α)} (hs : s.Infinite) : (⋃₀ s).Infinite
Set.Finite.sUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {s : Set (Set α)} (hs : s.Finite) (H : ∀ t ∈ s, t.Finite) : (⋃₀ s).Finite
Finite.Set.finite_sUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {s : Set (Set α)} [Finite ↑s] [H : ∀ (t : ↑s), Finite ↑↑t] : Finite ↑(⋃₀ s)
Set.fintypesUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} [DecidableEq α] {s : Set (Set α)} [Fintype ↑s] [H : (t : ↑s) → Fintype ↑↑t] : Fintype ↑(⋃₀ s)
DirectedOn.exists_mem_subset_of_finite_of_subset_sUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u_1} {c : Set (Set α)} (hn : c.Nonempty) (hc : DirectedOn (fun x1 x2 => x1 ⊆ x2) c) {s : Set α} (hs : s.Finite) (hsc : s ⊆ ⋃₀ c) : ∃ t ∈ c, s ⊆ t
Set.Countable.sUnion 📋 Mathlib.Data.Set.Countable
{α : Type u} {s : Set (Set α)} (hs : s.Countable) : (∀ a ∈ s, a.Countable) → (⋃₀ s).Countable
Set.Countable.sUnion_iff 📋 Mathlib.Data.Set.Countable
{α : Type u} {s : Set (Set α)} (hs : s.Countable) : (⋃₀ s).Countable ↔ ∀ a ∈ s, a.Countable
Set.exists_seq_cover_iff_countable 📋 Mathlib.Data.Set.Countable
{α : Type u} {p : Set α → Prop} (h : ∃ s, p s) : (∃ s, (∀ (n : ℕ), p (s n)) ∧ ⋃ n, s n = Set.univ) ↔ ∃ S, S.Countable ∧ (∀ s ∈ S, p s) ∧ ⋃₀ S = Set.univ
small_sUnion 📋 Mathlib.Logic.Small.Set
{α : Type u1} (s : Set (Set α)) [Small.{u, u1} ↑s] [∀ (t : ↑s), Small.{u, u1} ↑↑t] : Small.{u, u1} ↑(⋃₀ s)
isLowerSet_sUnion 📋 Mathlib.Order.UpperLower.Basic
{α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsLowerSet s) : IsLowerSet (⋃₀ S)
isUpperSet_sUnion 📋 Mathlib.Order.UpperLower.Basic
{α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsUpperSet s) : IsUpperSet (⋃₀ S)
ChainClosure.union 📋 Mathlib.Order.CompleteLattice.Chain
{α : Type u_1} {r : α → α → Prop} {s : Set (Set α)} : (∀ a ∈ s, ChainClosure r a) → ChainClosure r (⋃₀ s)
Cardinal.mk_sUnion_le 📋 Mathlib.SetTheory.Cardinal.Basic
{α : Type u} (A : Set (Set α)) : Cardinal.mk ↑(⋃₀ A) ≤ Cardinal.mk ↑A * ⨆ s, Cardinal.mk ↑↑s
NonUnitalSubsemiring.closure_sUnion 📋 Mathlib.RingTheory.NonUnitalSubsemiring.Basic
{R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (Set R)) : NonUnitalSubsemiring.closure (⋃₀ s) = ⨆ t ∈ s, NonUnitalSubsemiring.closure t
Subsemiring.closure_sUnion 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
{R : Type u} [NonAssocSemiring R] (s : Set (Set R)) : Subsemiring.closure (⋃₀ s) = ⨆ t ∈ s, Subsemiring.closure t
NonUnitalSubring.closure_sUnion 📋 Mathlib.RingTheory.NonUnitalSubring.Basic
{R : Type u} [NonUnitalNonAssocRing R] (s : Set (Set R)) : NonUnitalSubring.closure (⋃₀ s) = ⨆ t ∈ s, NonUnitalSubring.closure t
Subring.closure_sUnion 📋 Mathlib.Algebra.Ring.Subring.Basic
{R : Type u} [NonAssocRing R] (s : Set (Set R)) : Subring.closure (⋃₀ s) = ⨆ t ∈ s, Subring.closure t
Submodule.span_sSup 📋 Mathlib.LinearAlgebra.Span.Defs
{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set (Set M)) : Submodule.span R (⋃₀ s) = sSup (Submodule.span R '' s)
forall_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{α : Type u_1} {S : Set (Set α)} {p : α → Prop} : (∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x
exists_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{α : Type u_1} {S : Set (Set α)} {p : α → Prop} : (∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x
sInf_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{β : Type u_2} [CompleteLattice β] (s : Set (Set β)) : sInf (⋃₀ s) = ⨅ t ∈ s, sInf t
sSup_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{β : Type u_2} [CompleteLattice β] (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t
iInf_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{α : Type u_1} {β : Type u_2} [CompleteLattice β] (S : Set (Set α)) (f : α → β) : ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x
iSup_sUnion 📋 Mathlib.Data.Set.Lattice.Order
{α : Type u_1} {β : Type u_2} [CompleteLattice β] (S : Set (Set α)) (f : α → β) : ⨆ x ∈ ⋃₀ S, f x = ⨆ s ∈ S, ⨆ x ∈ s, f x
iSupIndep_sUnion_of_directed 📋 Mathlib.Order.CompactlyGenerated.Basic
{α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {s : Set (Set α)} (hs : DirectedOn (fun x1 x2 => x1 ⊆ x2) s) (h : ∀ a ∈ s, sSupIndep a) : sSupIndep (⋃₀ s)
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
partialSups_eq_sUnion_image 📋 Mathlib.Order.PartialSups
{α : Type u_1} (s : ℕ → Set α) (n : ℕ) : (partialSups s) n = ⋃₀ ↑(Finset.image s (Finset.range (n + 1)))
Set.sUnion_finite_eq_univ 📋 Mathlib.Data.Set.Finite.Lemmas
{X : Type u_1} : ⋃₀ {s | s.Finite} = Set.univ
isCofinal_of_isCofinal_sUnion 📋 Mathlib.SetTheory.Cardinal.Cofinality.Basic
{α : Type u_1} [LinearOrder α] {s : Set (Set α)} (h₁ : IsCofinal (⋃₀ s)) (h₂ : Cardinal.mk ↑s < Order.cof α) : ∃ x ∈ s, IsCofinal x
DirSupInacc.sUnion 📋 Mathlib.Order.DirSupClosed
{α : Type u_1} [Preorder α] {s : Set (Set α)} (hs : ∀ x ∈ s, DirSupInacc x) : DirSupInacc (⋃₀ s)
DirSupInaccOn.sUnion 📋 Mathlib.Order.DirSupClosed
{α : Type u_1} {D : Set (Set α)} [Preorder α] {s : Set (Set α)} (hs : ∀ x ∈ s, DirSupInaccOn D x) : DirSupInaccOn D (⋃₀ s)
linearIndepOn_sUnion_of_directed 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set (Set ι)} (hs : DirectedOn (fun x1 x2 => x1 ⊆ x2) s) (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s)
SetRel.image_sUnion 📋 Mathlib.Basic.Rel
{α : Type u_1} {β : Type u_2} (R : SetRel α β) (S : Set (Set α)) : R.image (⋃₀ S) = ⋃ s ∈ S, R.image s
SetRel.preimage_sUnion 📋 Mathlib.Basic.Rel
{α : Type u_1} {β : Type u_2} (R : SetRel α β) (T : Set (Set β)) : R.preimage (⋃₀ T) = ⋃ t ∈ T, R.preimage t
SetRel.comp_sUnion 📋 Mathlib.Basic.Rel
{α : Type u_1} {β : Type u_2} {γ : Type u_3} (R : SetRel α β) (𝒮 : Set (SetRel β γ)) : R.comp (⋃₀ 𝒮) = ⋃ S ∈ 𝒮, R.comp S
SetRel.sUnion_comp 📋 Mathlib.Basic.Rel
{α : Type u_1} {β : Type u_2} {γ : Type u_3} (ℛ : Set (SetRel α β)) (S : SetRel β γ) : SetRel.comp (⋃₀ ℛ) S = ⋃ R ∈ ℛ, R.comp S
Algebra.sSup_def 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Set (Subalgebra R A)) : sSup S = Algebra.adjoin R (⋃₀ (SetLike.coe '' S))
Subfield.closure_sUnion 📋 Mathlib.Algebra.Field.Subfield.Basic
{K : Type u} [DivisionRing K] (s : Set (Set K)) : Subfield.closure (⋃₀ s) = ⨆ t ∈ s, Subfield.closure t
isOpen_sUnion 📋 Mathlib.Topology.Defs.Basic
{X : Type u} [TopologicalSpace X] {s : Set (Set X)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s)
TopologicalSpace.isOpen_sUnion 📋 Mathlib.Topology.Defs.Basic
{X : Type u} [self : TopologicalSpace X] (s : Set (Set X)) : (∀ t ∈ s, TopologicalSpace.IsOpen t) → TopologicalSpace.IsOpen (⋃₀ s)
TopologicalSpace.mk 📋 Mathlib.Topology.Defs.Basic
{X : Type u} (IsOpen : Set X → Prop) (isOpen_univ : IsOpen Set.univ) (isOpen_inter : ∀ (s t : Set X), IsOpen s → IsOpen t → IsOpen (s ∩ t)) (isOpen_sUnion : ∀ (s : Set (Set X)), (∀ t ∈ s, IsOpen t) → IsOpen (⋃₀ s)) : TopologicalSpace X
Set.Finite.isClosed_sUnion 📋 Mathlib.Topology.Basic
{X : Type u} [TopologicalSpace X] {s : Set (Set X)} (hs : s.Finite) (h : ∀ t ∈ s, IsClosed t) : IsClosed (⋃₀ s)
isOpen_mk 📋 Mathlib.Topology.Basic
{X : Type u} {s : Set X} {p : Set X → Prop} {h₁ : p Set.univ} {h₂ : ∀ (s t : Set X), p s → p t → p (s ∩ t)} {h₃ : ∀ (s : Set (Set X)), (∀ t ∈ s, p t) → p (⋃₀ s)} : IsOpen s ↔ p s
Set.Finite.closure_sUnion 📋 Mathlib.Topology.Closure
{X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) : closure (⋃₀ S) = ⋃ s ∈ S, closure s
TopologicalSpace.GenerateOpen.sUnion 📋 Mathlib.Topology.Order
{α : Type u} {g : Set (Set α)} (S : Set (Set α)) : (∀ s ∈ S, TopologicalSpace.GenerateOpen g s) → TopologicalSpace.GenerateOpen g (⋃₀ S)
generateFrom_sUnion 📋 Mathlib.Topology.Order
{α : Type u} {S : Set (Set (Set α))} : TopologicalSpace.generateFrom (⋃₀ S) = ⨅ s ∈ S, TopologicalSpace.generateFrom s
prod_generateFrom_generateFrom_eq 📋 Mathlib.Topology.Constructions.SumProd
{X : Type u_5} {Y : Type u_6} {s : Set (Set X)} {t : Set (Set Y)} (hs : ⋃₀ s = Set.univ) (ht : ⋃₀ t = Set.univ) : instTopologicalSpaceProd = TopologicalSpace.generateFrom (Set.image2 (fun x1 x2 => x1 ×ˢ x2) s t)
pi_generateFrom_eq_finite 📋 Mathlib.Topology.Constructions
{ι : Type u_2} {X : ι → Type u_6} {g : (a : ι) → Set (Set (X a))} [Finite ι] (hg : ∀ (a : ι), ⋃₀ g a = Set.univ) : Pi.topologicalSpace = TopologicalSpace.generateFrom {t | ∃ s, (∀ (a : ι), s a ∈ g a) ∧ t = Set.univ.pi s}
Pi.induced_domRestrict_sUnion 📋 Mathlib.Topology.Constructions
{ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (𝔖 : Set (Set ι)) : TopologicalSpace.induced (⋃₀ 𝔖).domRestrict Pi.topologicalSpace = ⨅ S ∈ 𝔖, TopologicalSpace.induced S.domRestrict Pi.topologicalSpace
Pi.induced_restrict_sUnion 📋 Mathlib.Topology.Constructions
{ι : Type u_2} {A : ι → Type u_3} [T : (i : ι) → TopologicalSpace (A i)] (𝔖 : Set (Set ι)) : TopologicalSpace.induced (⋃₀ 𝔖).domRestrict Pi.topologicalSpace = ⨅ S ∈ 𝔖, TopologicalSpace.induced S.domRestrict Pi.topologicalSpace
nhdsWithin_sUnion 📋 Mathlib.Topology.NhdsWithin
{α : Type u_1} [TopologicalSpace α] {S : Set (Set α)} (hS : S.Finite) (a : α) : nhdsWithin a (⋃₀ S) = ⨆ s ∈ S, nhdsWithin a s
TopologicalSpace.IsTopologicalBasis.sUnion_eq 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (self : TopologicalSpace.IsTopologicalBasis s) : ⋃₀ s = Set.univ
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (ou : IsOpen u) : ∃ S ⊆ B, u = ⋃₀ S
TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} : IsOpen u ↔ ∃ S ⊆ B, u = ⋃₀ S
TopologicalSpace.IsTopologicalBasis.finite_sUnion 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) : TopologicalSpace.IsTopologicalBasis (Set.sUnion '' {f | f.Finite ∧ f ⊆ B})
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion' 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} (hB : TopologicalSpace.IsTopologicalBasis B) {u : Set α} (ou : IsOpen u) : u = ⋃₀ {s | s ∈ B ∧ s ⊆ u}
TopologicalSpace.isOpen_sUnion_countable 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] [SecondCountableTopology α] (S : Set (Set α)) (H : ∀ s ∈ S, IsOpen s) : ∃ T, T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S
TopologicalSpace.IsTopologicalBasis.isOpen_induction 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} {P : Set α → Prop} (hB : TopologicalSpace.IsTopologicalBasis B) (basis : ∀ b ∈ B, P b) (sUnion : ∀ (S : Set (Set α)), (∀ s ∈ S, P s) → P (⋃₀ S)) {s : Set α} (hs : IsOpen s) : P s
TopologicalSpace.IsTopologicalBasis.mk 📋 Mathlib.Topology.Bases
{α : Type u} [t : TopologicalSpace α] {s : Set (Set α)} (exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂) (sUnion_eq : ⋃₀ s = Set.univ) (eq_generateFrom : t = TopologicalSpace.generateFrom s) : TopologicalSpace.IsTopologicalBasis s
Bornology.sUnion_bounded_univ 📋 Mathlib.Topology.Bornology.Basic
{α : Type u_2} {x✝ : Bornology α} : ⋃₀ {s | Bornology.IsBounded s} = Set.univ
Bornology.isBounded_sUnion 📋 Mathlib.Topology.Bornology.Basic
{α : Type u_2} [Bornology α] {S : Set (Set α)} (hs : S.Finite) : Bornology.IsBounded (⋃₀ S) ↔ ∀ s ∈ S, Bornology.IsBounded s
Bornology.ofBounded' 📋 Mathlib.Topology.Bornology.Basic
{α : Type u_4} (B : Set (Set α)) (empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B) (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (sUnion_univ : ⋃₀ B = Set.univ) : Bornology α
Bornology.ofBounded'_cobounded 📋 Mathlib.Topology.Bornology.Basic
{α : Type u_4} (B : Set (Set α)) (empty_mem : ∅ ∈ B) (subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B) (union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B) (sUnion_univ : ⋃₀ B = Set.univ) : Bornology.cobounded α = Filter.comk (fun x => x ∈ B) empty_mem subset_mem union_mem
Ultrafilter.finite_sUnion_mem_iff 📋 Mathlib.Order.Filter.Ultrafilter.Basic
{α : Type u} {f : Ultrafilter α} {s : Set (Set α)} (hs : s.Finite) : ⋃₀ s ∈ f ↔ ∃ t ∈ s, t ∈ f
Set.sUnion_isCompact_eq_univ 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] : ⋃₀ {s | IsCompact s} = Set.univ
Set.Finite.isCompact_sUnion 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsCompact s) : IsCompact (⋃₀ S)
compactSpace_generateFrom 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [T : TopologicalSpace X] {S : Set (Set X)} (hTS : T = TopologicalSpace.generateFrom S) (h : ∀ P ⊆ S, ⋃₀ P = Set.univ → ∃ Q ⊆ P, Q.Finite ∧ ⋃₀ Q = Set.univ) : CompactSpace X
isCompact_generateFrom 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [T : TopologicalSpace X] {S : Set (Set X)} (hTS : T = TopologicalSpace.generateFrom S) {s : Set X} (h : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q) : IsCompact s
IsCompact.elim_directedOn_cover 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (U : Set (Set X)) (hUo : ∀ u ∈ U, IsOpen u) (hsU : s ⊆ ⋃₀ U) (hdU : DirectedOn (fun x1 x2 => x1 ⊆ x2) U) (hU : U.Nonempty) : ∃ u ∈ U, s ⊆ u
isSigmaCompact_sUnion_of_isCompact 📋 Mathlib.Topology.Compactness.SigmaCompact
{X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (hc : S.Countable) (hcomp : ∀ s ∈ S, IsCompact s) : IsSigmaCompact (⋃₀ S)
SigmaCompactSpace.of_countable 📋 Mathlib.Topology.Compactness.SigmaCompact
{X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (Hc : S.Countable) (Hcomp : ∀ s ∈ S, IsCompact s) (HU : ⋃₀ S = Set.univ) : SigmaCompactSpace X
isSigmaCompact_sUnion 📋 Mathlib.Topology.Compactness.SigmaCompact
{X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (hc : S.Countable) (hcomp : ∀ (s : ↑S), IsSigmaCompact ↑s) : IsSigmaCompact (⋃₀ S)
lowerClosure_sUnion 📋 Mathlib.Order.UpperLower.Closure
{α : Type u_1} [Preorder α] (S : Set (Set α)) : lowerClosure (⋃₀ S) = ⨆ s ∈ S, lowerClosure s
upperClosure_sUnion 📋 Mathlib.Order.UpperLower.Closure
{α : Type u_1} [Preorder α] (S : Set (Set α)) : upperClosure (⋃₀ S) = ⨅ s ∈ S, upperClosure s
stableUnderGeneralization_sUnion 📋 Mathlib.Topology.Inseparable
{X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderGeneralization s) : StableUnderGeneralization (⋃₀ S)
stableUnderSpecialization_sUnion 📋 Mathlib.Topology.Inseparable
{X : Type u_1} [TopologicalSpace X] (S : Set (Set X)) (H : ∀ s ∈ S, StableUnderSpecialization s) : StableUnderSpecialization (⋃₀ S)
stableUnderSpecialization_iff_exists_sUnion_eq 📋 Mathlib.Topology.Inseparable
{X : Type u_1} [TopologicalSpace X] {s : Set X} : StableUnderSpecialization s ↔ ∃ S, (∀ s ∈ S, IsClosed s) ∧ ⋃₀ S = s
sUnion_irreducibleComponents 📋 Mathlib.Topology.Irreducible
{X : Type u_1} [TopologicalSpace X] : ⋃₀ irreducibleComponents X = Set.univ
mem_of_subset_sUnion_irreducibleComponents 📋 Mathlib.Topology.Irreducible
{X : Type u_1} [TopologicalSpace X] (Z : Set X) (hZ : Z ∈ irreducibleComponents X) (S : Set (Set X)) (hS : S.Finite) (hSα : S ⊆ irreducibleComponents X) (hZS : Z ⊆ ⋃₀ S) : Z ∈ S
closure_sUnion_irreducibleComponents_sdiff_singleton 📋 Mathlib.Topology.Irreducible
{X : Type u_1} [TopologicalSpace X] (hX : (irreducibleComponents X).Finite) (Z : Set X) (hZ : Z ∈ irreducibleComponents X) : closure (⋃₀ (irreducibleComponents X \ {Z}))ᶜ = Z
isIrreducible_iff_sUnion_isClosed 📋 Mathlib.Topology.Irreducible
{X : Type u_1} [TopologicalSpace X] {s : Set X} : IsIrreducible s ↔ ∀ (t : Finset (Set X)), (∀ z ∈ t, IsClosed z) → s ⊆ ⋃₀ ↑t → ∃ z ∈ t, s ⊆ z
IsPreconnected.sUnion_directed 📋 Mathlib.Topology.Connected.Basic
{α : Type u} [TopologicalSpace α] {S : Set (Set α)} (K : DirectedOn (fun x1 x2 => x1 ⊆ x2) S) (H : ∀ s ∈ S, IsPreconnected s) : IsPreconnected (⋃₀ S)
isPreconnected_sUnion 📋 Mathlib.Topology.Connected.Basic
{α : Type u} [TopologicalSpace α] (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s) (H2 : ∀ s ∈ c, IsPreconnected s) : IsPreconnected (⋃₀ c)
Set.preimage_val_sUnion 📋 Mathlib.Data.Set.Subset
{α : Type u_2} {A : Set α} {S : Set (Set α)} : Subtype.val ⁻¹' ⋃₀ S = ⋃₀ {x | ∃ B ∈ S, Subtype.val ⁻¹' B = x}
Set.image_val_sUnion 📋 Mathlib.Data.Set.Subset
{α : Type u_2} {A : Set α} {T : Set (Set ↑A)} : Subtype.val '' ⋃₀ T = ⋃₀ {x | ∃ B ∈ T, Subtype.val '' B = x}
isConnected_iff_sUnion_disjoint_open 📋 Mathlib.Topology.Connected.Clopen
{α : Type u} [TopologicalSpace α] {s : Set α} : IsConnected s ↔ ∀ (U : Finset (Set α)), (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) → (∀ u ∈ U, IsOpen u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u
Filter.ofCountableUnion 📋 Mathlib.Order.Filter.CountableInter
{α : Type u_2} (l : Set (Set α)) (hUnion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l) (hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l) : Filter α
Filter.countableInter_ofCountableUnion 📋 Mathlib.Order.Filter.CountableInter
{α : Type u_2} (l : Set (Set α)) (h₁ : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l) (h₂ : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l) : CountableInterFilter (Filter.ofCountableUnion l h₁ h₂)
Filter.mem_ofCountableUnion 📋 Mathlib.Order.Filter.CountableInter
{α : Type u_2} {l : Set (Set α)} {hunion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l} {hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l} {s : Set α} : s ∈ Filter.ofCountableUnion l hunion hmono ↔ sᶜ ∈ l
Set.Countable.isLindelof_sUnion 📋 Mathlib.Topology.Compactness.Lindelof
{X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Countable) (hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S)
Set.Finite.isLindelof_sUnion 📋 Mathlib.Topology.Compactness.Lindelof
{X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S)
IsLindelof.induction_on 📋 Mathlib.Topology.Compactness.Lindelof
{X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {p : Set X → Prop} (hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s) (hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S)) (hnhds : ∀ x ∈ s, ∃ t ∈ nhdsWithin x s, p t) : p s
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion_of_closure_subset 📋 Mathlib.Topology.Separation.Regular
{X : Type u_1} [TopologicalSpace X] [RegularSpace X] {B : Set (Set X)} (hB : TopologicalSpace.IsTopologicalBasis B) {U : Set X} (hU : IsOpen U) : U = ⋃₀ {v | v ∈ B ∧ closure v ⊆ U}
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion_closure 📋 Mathlib.Topology.Separation.Regular
{X : Type u_1} [TopologicalSpace X] [RegularSpace X] {B : Set (Set X)} (hB : TopologicalSpace.IsTopologicalBasis B) {U : Set X} (hU : IsOpen U) : U = ⋃₀ {v | ∃ u ∈ B, closure u ⊆ U ∧ v = closure u}
IsGδ.sUnion 📋 Mathlib.Topology.GDelta.Basic
{X : Type u_1} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s) : IsGδ (⋃₀ S)
isMeagre_iff_countable_union_isNowhereDense 📋 Mathlib.Topology.GDelta.Basic
{X : Type u_1} [TopologicalSpace X] {s : Set X} : IsMeagre s ↔ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
dense_sUnion_interior_of_closed 📋 Mathlib.Topology.Baire.Lemmas
{X : Type u_1} [TopologicalSpace X] [BaireSpace X] {S : Set (Set X)} (hc : ∀ s ∈ S, IsClosed s) (hS : S.Countable) (hU : ⋃₀ S = Set.univ) : Dense (⋃ s ∈ S, interior s)
IsGδ.dense_sUnion_interior_of_closed 📋 Mathlib.Topology.Baire.Lemmas
{X : Type u_1} [TopologicalSpace X] [BaireSpace X] {T : Set (Set X)} {s : Set X} (hs : IsGδ s) (hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T) : Dense (⋃ t ∈ T, interior t)
totallyBounded_sUnion 📋 Mathlib.Topology.UniformSpace.Cauchy
{α : Type u} [uniformSpace : UniformSpace α] {S : Set (Set α)} (hS : S.Finite) : TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s
lebesgue_number_lemma_sUnion 📋 Mathlib.Topology.UniformSpace.Compact
{α : Type ua} [UniformSpace α] {K : Set α} {S : Set (Set α)} (hK : IsCompact K) (hopen : ∀ s ∈ S, IsOpen s) (hcover : K ⊆ ⋃₀ S) : ∃ V ∈ uniformity α, ∀ x ∈ K, ∃ s ∈ S, UniformSpace.ball x V ⊆ s
Pi.uniformSpace_comap_restrict_sUnion 📋 Mathlib.Topology.UniformSpace.Pi
{ι : Type u_1} (α : ι → Type u) [U : (i : ι) → UniformSpace (α i)] (𝔖 : Set (Set ι)) : UniformSpace.comap (⋃₀ 𝔖).domRestrict (Pi.uniformSpace fun i => α ↑i) = ⨅ S ∈ 𝔖, UniformSpace.comap S.domRestrict (Pi.uniformSpace fun i => α ↑i)
tendstoLocallyUniformlyOn_sUnion 📋 Mathlib.Topology.UniformSpace.LocallyUniformConvergence
{α : Type u_1} {β : Type u_2} {ι : Type u_4} [TopologicalSpace α] [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} (S : Set (Set α)) (hS : ∀ s ∈ S, IsOpen s) (h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s) : TendstoLocallyUniformlyOn F f p (⋃₀ S)
UniformOnFun.t2Space_of_covering 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [T2Space β] (h : ⋃₀ 𝔖 = Set.univ) : T2Space (UniformOnFun α β 𝔖)
UniformOnFun.uniformContinuous_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} (h : ⋃₀ 𝔖 = Set.univ) : UniformContinuous ⇑(UniformOnFun.toFun 𝔖)
UniformOnFun.uniformContinuous_eval 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} (h : ⋃₀ 𝔖 = Set.univ) (x : α) : UniformContinuous (Function.eval x ∘ ⇑(UniformOnFun.toFun 𝔖))
UniformOnFun.uniformContinuous_eval_of_mem_sUnion 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 : Set (Set α)) {x : α} (hx : x ∈ ⋃₀ 𝔖) : UniformContinuous (Function.eval x ∘ ⇑(UniformOnFun.toFun 𝔖))
UniformOnFun.uniformContinuous_restrict_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} : UniformContinuous ((⋃₀ 𝔖).domRestrict ∘ ⇑(UniformOnFun.toFun 𝔖))
UniformOnFun.uniformContinuous_ofFun_toFun 📋 Mathlib.Topology.UniformSpace.UniformConvergenceTopology
{α : Type u_1} (β : Type u_2) [UniformSpace β] (𝔖 𝔗 : Set (Set α)) (h : ∀ s ∈ 𝔖, ∃ T ⊆ 𝔗, T.Finite ∧ s ⊆ ⋃₀ T) : UniformContinuous (⇑(UniformOnFun.ofFun 𝔗) ∘ ⇑(UniformOnFun.toFun 𝔖))
eq_sUnion_finset_of_isTopologicalBasis_of_isCompact_open 📋 Mathlib.Topology.Compactness.Bases
{X : Type u_1} [TopologicalSpace X] (b : Set (Set X)) (hb : TopologicalSpace.IsTopologicalBasis b) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) : ∃ s, U = ⋃₀ (Subtype.val '' ↑s)
TopologicalSpace.Closeds.coe_sSup 📋 Mathlib.Topology.Sets.Closeds
{α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)} : ↑(sSup S) = closure (⋃₀ (SetLike.coe '' S))
TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible 📋 Mathlib.Topology.NoetherianSpace
{α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] {s : Set α} (hs : IsClosed s) : ∃ S, S.Finite ∧ (∀ t ∈ S, IsClosed t) ∧ (∀ t ∈ S, IsIrreducible t) ∧ s = ⋃₀ S
Order.isIdeal_sUnion_of_directedOn 📋 Mathlib.Order.Ideal
{P : Type u_1} [LE P] {C : Set (Set P)} (hidl : ∀ I ∈ C, Order.IsIdeal I) (hD : DirectedOn (fun x1 x2 => x1 ⊆ x2) C) (hNe : C.Nonempty) : Order.IsIdeal (⋃₀ C)
Order.isIdeal_sUnion_of_isChain 📋 Mathlib.Order.Ideal
{P : Type u_1} [LE P] {C : Set (Set P)} (hidl : ∀ I ∈ C, Order.IsIdeal I) (hC : IsChain (fun x1 x2 => x1 ⊆ x2) C) (hNe : C.Nonempty) : Order.IsIdeal (⋃₀ C)
IsRetrocompact.sUnion 📋 Mathlib.Topology.Constructible
{X : Type u_2} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsRetrocompact s) : IsRetrocompact (⋃₀ S)
Topology.IsConstructible.sUnion 📋 Mathlib.Topology.Constructible
{X : Type u_2} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, Topology.IsConstructible s) : Topology.IsConstructible (⋃₀ S)
Topology.IsLocallyConstructible.sUnion 📋 Mathlib.Topology.Constructible
{X : Type u_2} [TopologicalSpace X] {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, Topology.IsLocallyConstructible s) : Topology.IsLocallyConstructible (⋃₀ S)
quasiSober_of_open_cover 📋 Mathlib.Topology.Sober
{α : Type u_1} [TopologicalSpace α] (S : Set (Set α)) (hS : ∀ (s : ↑S), IsOpen ↑s) [∀ (s : ↑S), QuasiSober ↑↑s] (hS' : ⋃₀ S = ⊤) : QuasiSober α
IntermediateField.sSup_def 📋 Mathlib.FieldTheory.IntermediateField.Adjoin.Defs
{F : Type u_1} [Field F] {E : Type u_2} [Field E] [Algebra F E] (S : Set (IntermediateField F E)) : sSup S = IntermediateField.adjoin F (⋃₀ (SetLike.coe '' S))
Matroid.IsBasis.isBasis_sUnion 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {I : Set α} {Xs : Set (Set α)} (hne : Xs.Nonempty) (h : ∀ X ∈ Xs, M.IsBasis I X) : M.IsBasis I (⋃₀ Xs)

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:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
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
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.
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 ce5dd8c