Loogle!

Result

Found 493 declarations mentioning Set.EqOn. Of these, only the first 200 are shown.

• Set.EqOn 📋 Mathlib.Data.Set.Operations
  {α : Type u} {β : Type v} (f₁ f₂ : α → β) (s : Set α) : Prop
• Set.graphOn_inj 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {s : Set α} {f g : α → β} : Set.graphOn f s = Set.graphOn g s ↔ Set.EqOn f g s
• Set.EqOn.left_of_eqOn_prodMap 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {s : Set α} {t : Set β} {f f' : α → γ} {g g' : β → δ} (h : Set.EqOn (Prod.map f g) (Prod.map f' g') (s ×ˢ t)) (ht : t.Nonempty) : Set.EqOn f f' s
• Set.EqOn.right_of_eqOn_prodMap 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {s : Set α} {t : Set β} {f f' : α → γ} {g g' : β → δ} (h : Set.EqOn (Prod.map f g) (Prod.map f' g') (s ×ˢ t)) (hs : s.Nonempty) : Set.EqOn g g' t
• Set.EqOn.prodMap 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {s : Set α} {t : Set β} {f f' : α → γ} {g g' : β → δ} (hf : Set.EqOn f f' s) (hg : Set.EqOn g g' t) : Set.EqOn (Prod.map f g) (Prod.map f' g') (s ×ˢ t)
• Set.eqOn_prodMap_iff 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f f' : α → γ} {g g' : β → δ} {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty) : Set.EqOn (Prod.map f g) (Prod.map f' g') (s ×ˢ t) ↔ Set.EqOn f f' s ∧ Set.EqOn g g' t
• Set.eqOn_prod_iff 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {γ : Type u_3} {δ : Type u_4} {s : Set α} {a b : α → γ × δ} : Set.EqOn a b s ↔ Set.EqOn (Prod.fst ∘ a) (Prod.fst ∘ b) s ∧ Set.EqOn (Prod.snd ∘ a) (Prod.snd ∘ b) s
• Set.domRestrict_eq_domRestrict_iff 📋 Mathlib.Data.Set.Restrict
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} : s.domRestrict f₁ = s.domRestrict f₂ ↔ Set.EqOn f₁ f₂ s
• Set.restrict_eq_restrict_iff 📋 Mathlib.Data.Set.Restrict
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} : s.domRestrict f₁ = s.domRestrict f₂ ↔ Set.EqOn f₁ f₂ s
• Set.eqOn_refl 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} (f : α → β) (s : Set α) : Set.EqOn f f s
• Set.eqOn_empty 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} (f₁ f₂ : α → β) : Set.EqOn f₁ f₂ ∅
• Set.eqOn_univ 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} (f₁ f₂ : α → β) : Set.EqOn f₁ f₂ Set.univ ↔ f₁ = f₂
• Set.EqOn.symm 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} (h : Set.EqOn f₁ f₂ s) : Set.EqOn f₂ f₁ s
• Set.eqOn_comm 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} : Set.EqOn f₁ f₂ s ↔ Set.EqOn f₂ f₁ s
• Set.EqOn.image_eq_self 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {s : Set α} {f : α → α} (h : Set.EqOn f id s) : f '' s = s
• Set.InjOn.congr 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} (h₁ : Set.InjOn f₁ s) (h : Set.EqOn f₁ f₂ s) : Set.InjOn f₂ s
• Set.EqOn.injOn_iff 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} (H : Set.EqOn f₁ f₂ s) : Set.InjOn f₁ s ↔ Set.InjOn f₂ s
• Set.eqOn_singleton 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {f₁ f₂ : α → β} {a : α} : Set.EqOn f₁ f₂ {a} ↔ f₁ a = f₂ a
• Set.LeftInvOn.eqOn 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {f' : β → α} (h : Set.LeftInvOn f' f s) : Set.EqOn (f' ∘ f) id s
• Set.RightInvOn.eqOn 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {t : Set β} {f : α → β} {f' : β → α} (h : Set.RightInvOn f' f t) : Set.EqOn (f ∘ f') id t
• Set.EqOn.image_eq 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} (heq : Set.EqOn f₁ f₂ s) : f₁ '' s = f₂ '' s
• Set.BijOn.congr 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (h₁ : Set.BijOn f₁ s t) (h : Set.EqOn f₁ f₂ s) : Set.BijOn f₂ s t
• Set.EqOn.trans 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ f₃ : α → β} (h₁ : Set.EqOn f₁ f₂ s) (h₂ : Set.EqOn f₂ f₃ s) : Set.EqOn f₁ f₃ s
• Set.LeftInvOn.congr_right 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} {f₁' : β → α} (h₁ : Set.LeftInvOn f₁' f₁ s) (heq : Set.EqOn f₁ f₂ s) : Set.LeftInvOn f₁' f₂ s
• Set.MapsTo.congr 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (h₁ : Set.MapsTo f₁ s t) (h : Set.EqOn f₁ f₂ s) : Set.MapsTo f₂ s t
• Set.RightInvOn.congr_left 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {t : Set β} {f : α → β} {f₁' f₂' : β → α} (h₁ : Set.RightInvOn f₁' f t) (heq : Set.EqOn f₁' f₂' t) : Set.RightInvOn f₂' f t
• Set.SurjOn.congr 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (h : Set.SurjOn f₁ s t) (H : Set.EqOn f₁ f₂ s) : Set.SurjOn f₂ s t
• Set.EqOn.bijOn_iff 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (H : Set.EqOn f₁ f₂ s) : Set.BijOn f₁ s t ↔ Set.BijOn f₂ s t
• Set.EqOn.eq_of_mem 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} {a : α} (h : Set.EqOn f₁ f₂ s) (ha : a ∈ s) : f₁ a = f₂ a
• Set.EqOn.mapsTo_iff 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (H : Set.EqOn f₁ f₂ s) : Set.MapsTo f₁ s t ↔ Set.MapsTo f₂ s t
• Set.EqOn.mono 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s₁ s₂ : Set α} {f₁ f₂ : α → β} (hs : s₁ ⊆ s₂) (hf : Set.EqOn f₁ f₂ s₂) : Set.EqOn f₁ f₂ s₁
• Set.EqOn.surjOn_iff 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} (h : Set.EqOn f₁ f₂ s) : Set.SurjOn f₁ s t ↔ Set.SurjOn f₂ s t
• Set.EqOn.comp_eq 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {ι : Sort u_6} {f : ι → α} {g₁ g₂ : α → β} : Set.EqOn g₁ g₂ (Set.range f) → g₁ ∘ f = g₂ ∘ f
• Set.EqOn.comp_left 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {f₁ f₂ : α → β} {g : β → γ} (h : Set.EqOn f₁ f₂ s) : Set.EqOn (g ∘ f₁) (g ∘ f₂) s
• Set.eqOn_range 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {ι : Sort u_6} {f : ι → α} {g₁ g₂ : α → β} : Set.EqOn g₁ g₂ (Set.range f) ↔ g₁ ∘ f = g₂ ∘ f
• Set.EqOn.union 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s₁ s₂ : Set α} {f₁ f₂ : α → β} (h₁ : Set.EqOn f₁ f₂ s₁) (h₂ : Set.EqOn f₁ f₂ s₂) : Set.EqOn f₁ f₂ (s₁ ∪ s₂)
• Set.eqOn_of_leftInvOn_of_rightInvOn 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f : α → β} {f₁' f₂' : β → α} (h₁ : Set.LeftInvOn f₁' f s) (h₂ : Set.RightInvOn f₂' f t) (h : Set.MapsTo f₂' t s) : Set.EqOn f₁' f₂' t
• Set.eqOn_union 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s₁ s₂ : Set α} {f₁ f₂ : α → β} : Set.EqOn f₁ f₂ (s₁ ∪ s₂) ↔ Set.EqOn f₁ f₂ s₁ ∧ Set.EqOn f₁ f₂ s₂
• Set.LeftInvOn.congr_left 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {f₁' f₂' : β → α} (h₁ : Set.LeftInvOn f₁' f s) {t : Set β} (h₁' : Set.MapsTo f s t) (heq : Set.EqOn f₁' f₂' t) : Set.LeftInvOn f₂' f s
• Set.RightInvOn.congr_right 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {f₁ f₂ : α → β} {f' : β → α} (h₁ : Set.RightInvOn f' f₁ t) (hg : Set.MapsTo f' t s) (heq : Set.EqOn f₁ f₂ s) : Set.RightInvOn f' f₂ t
• Set.eqOn_comp_right_iff 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {f : α → β} {g₁ g₂ : β → γ} : Set.EqOn (g₁ ∘ f) (g₂ ∘ f) s ↔ Set.EqOn g₁ g₂ (f '' s)
• Set.EqOn.cancel_right 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {f : α → β} {g₁ g₂ : β → γ} (hf : Set.EqOn (g₁ ∘ f) (g₂ ∘ f) s) (hf' : Set.SurjOn f s t) : Set.EqOn g₁ g₂ t
• Set.EqOn.comp_right 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {f : α → β} {g₁ g₂ : β → γ} (hg : Set.EqOn g₁ g₂ t) (hf : Set.MapsTo f s t) : Set.EqOn (g₁ ∘ f) (g₂ ∘ f) s
• Set.EqOn.inter_preimage_eq 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} (heq : Set.EqOn f₁ f₂ s) (t : Set β) : s ∩ f₁ ⁻¹' t = s ∩ f₂ ⁻¹' t
• Set.EqOn.comp_left₂ 📋 Mathlib.Data.Set.Function
  {α : Type u_6} {β : Type u_7} {δ : Type u_8} {γ : Type u_9} {op : α → β → δ} {a₁ a₂ : γ → α} {b₁ b₂ : γ → β} {s : Set γ} (ha : Set.EqOn a₁ a₂ s) (hb : Set.EqOn b₁ b₂ s) : Set.EqOn (fun x => op (a₁ x) (b₁ x)) (fun x => op (a₂ x) (b₂ x)) s
• Set.SurjOn.cancel_right 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {f : α → β} {g₁ g₂ : β → γ} (hf : Set.SurjOn f s t) (hf' : Set.MapsTo f s t) : Set.EqOn (g₁ ∘ f) (g₂ ∘ f) s ↔ Set.EqOn g₁ g₂ t
• Set.EqOn.cancel_left 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {f₁ f₂ : α → β} {g : β → γ} (h : Set.EqOn (g ∘ f₁) (g ∘ f₂) s) (hg : Set.InjOn g t) (hf₁ : Set.MapsTo f₁ s t) (hf₂ : Set.MapsTo f₂ s t) : Set.EqOn f₁ f₂ s
• Set.InjOn.cancel_left 📋 Mathlib.Data.Set.Function
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Set α} {t : Set β} {f₁ f₂ : α → β} {g : β → γ} (hg : Set.InjOn g t) (hf₁ : Set.MapsTo f₁ s t) (hf₂ : Set.MapsTo f₂ s t) : Set.EqOn (g ∘ f₁) (g ∘ f₂) s ↔ Set.EqOn f₁ f₂ s
• Set.piecewise_eqOn 📋 Mathlib.Data.Set.Piecewise
  {α : Type u_1} {β : Type u_2} (s : Set α) [(j : α) → Decidable (j ∈ s)] (f g : α → β) : Set.EqOn (s.piecewise f g) f s
• Set.piecewise_eqOn_compl 📋 Mathlib.Data.Set.Piecewise
  {α : Type u_1} {β : Type u_2} (s : Set α) [(j : α) → Decidable (j ∈ s)] (f g : α → β) : Set.EqOn (s.piecewise f g) g sᶜ
• Set.EqOn.piecewise_ite 📋 Mathlib.Data.Set.Piecewise
  {α : Type u_1} {β : Type u_2} (s : Set α) [(j : α) → Decidable (j ∈ s)] {f f' g : α → β} {t t' : Set α} (h : Set.EqOn f g t) (h' : Set.EqOn f' g t') : Set.EqOn (s.piecewise f f') g (s.ite t t')
• Set.eqOn_piecewise 📋 Mathlib.Data.Set.Piecewise
  {α : Type u_1} {β : Type u_2} (s : Set α) [(j : α) → Decidable (j ∈ s)] {f f' g : α → β} {t : Set α} : Set.EqOn (s.piecewise f f') g t ↔ Set.EqOn f g (t ∩ s) ∧ Set.EqOn f' g (t ∩ sᶜ)
• Set.EqOn.piecewise_ite' 📋 Mathlib.Data.Set.Piecewise
  {α : Type u_1} {β : Type u_2} (s : Set α) [(j : α) → Decidable (j ∈ s)] {f f' g : α → β} {t t' : Set α} (h : Set.EqOn f g (t ∩ s)) (h' : Set.EqOn f' g (t' ∩ sᶜ)) : Set.EqOn (s.piecewise f f') g (s.ite t t')
• AntitoneOn.congr 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h₁ : AntitoneOn f₁ s) (h : Set.EqOn f₁ f₂ s) : AntitoneOn f₂ s
• MonotoneOn.congr 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h₁ : MonotoneOn f₁ s) (h : Set.EqOn f₁ f₂ s) : MonotoneOn f₂ s
• StrictAntiOn.congr 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h₁ : StrictAntiOn f₁ s) (h : Set.EqOn f₁ f₂ s) : StrictAntiOn f₂ s
• StrictMonoOn.congr 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h₁ : StrictMonoOn f₁ s) (h : Set.EqOn f₁ f₂ s) : StrictMonoOn f₂ s
• Set.EqOn.congr_antitoneOn 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h : Set.EqOn f₁ f₂ s) : AntitoneOn f₁ s ↔ AntitoneOn f₂ s
• Set.EqOn.congr_monotoneOn 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h : Set.EqOn f₁ f₂ s) : MonotoneOn f₁ s ↔ MonotoneOn f₂ s
• Set.EqOn.congr_strictAntiOn 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h : Set.EqOn f₁ f₂ s) : StrictAntiOn f₁ s ↔ StrictAntiOn f₂ s
• Set.EqOn.congr_strictMonoOn 📋 Mathlib.Data.Set.Monotone
  {α : Type u_1} {β : Type u_2} {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β] (h : Set.EqOn f₁ f₂ s) : StrictMonoOn f₁ s ↔ StrictMonoOn f₂ s
• Function.disjoint_mulSupport_iff 📋 Mathlib.Algebra.Notation.Support
  {ι : Type u_1} {M : Type u_3} [One M] {f : ι → M} {s : Set ι} : Disjoint s (Function.mulSupport f) ↔ Set.EqOn f 1 s
• Function.disjoint_support_iff 📋 Mathlib.Algebra.Notation.Support
  {ι : Type u_1} {M : Type u_3} [Zero M] {f : ι → M} {s : Set ι} : Disjoint s (Function.support f) ↔ Set.EqOn f 0 s
• Function.mulSupport_disjoint_iff 📋 Mathlib.Algebra.Notation.Support
  {ι : Type u_1} {M : Type u_3} [One M] {f : ι → M} {s : Set ι} : Disjoint (Function.mulSupport f) s ↔ Set.EqOn f 1 s
• Function.support_disjoint_iff 📋 Mathlib.Algebra.Notation.Support
  {ι : Type u_1} {M : Type u_3} [Zero M] {f : ι → M} {s : Set ι} : Disjoint (Function.support f) s ↔ Set.EqOn f 0 s
• Finset.image_congr 📋 Mathlib.Data.Finset.Image
  {α : Type u_1} {β : Type u_2} [DecidableEq β] {f g : α → β} {s : Finset α} (h : Set.EqOn f g ↑s) : Finset.image f s = Finset.image g s
• AddHom.eq_of_eqOn_top 📋 Mathlib.Algebra.Group.Subsemigroup.Defs
  {M : Type u_1} {N : Type u_2} [Add M] [Add N] {f g : M →ₙ+ N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• MulHom.eq_of_eqOn_top 📋 Mathlib.Algebra.Group.Subsemigroup.Defs
  {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {f g : M →ₙ* N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• AddMonoidHom.eq_of_eqOn_topM 📋 Mathlib.Algebra.Group.Submonoid.Defs
  {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {f g : M →+ N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• MonoidHom.eq_of_eqOn_topM 📋 Mathlib.Algebra.Group.Submonoid.Defs
  {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {f g : M →* N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• AddHom.eq_of_eqOn_dense 📋 Mathlib.Algebra.Group.Subsemigroup.Basic
  {M : Type u_1} {N : Type u_2} [Add M] [Add N] {s : Set M} (hs : AddSubsemigroup.closure s = ⊤) {f g : M →ₙ+ N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• MulHom.eq_of_eqOn_dense 📋 Mathlib.Algebra.Group.Subsemigroup.Basic
  {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {s : Set M} (hs : Subsemigroup.closure s = ⊤) {f g : M →ₙ* N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• AddHom.eqOn_closure 📋 Mathlib.Algebra.Group.Subsemigroup.Basic
  {M : Type u_1} {N : Type u_2} [Add M] [Add N] {f g : M →ₙ+ N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(AddSubsemigroup.closure s)
• MulHom.eqOn_closure 📋 Mathlib.Algebra.Group.Subsemigroup.Basic
  {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {f g : M →ₙ* N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subsemigroup.closure s)
• AddMonoidHom.eq_of_eqOn_denseM 📋 Mathlib.Algebra.Group.Submonoid.Basic
  {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {s : Set M} (hs : AddSubmonoid.closure s = ⊤) {f g : M →+ N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• MonoidHom.eq_of_eqOn_denseM 📋 Mathlib.Algebra.Group.Submonoid.Basic
  {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {s : Set M} (hs : Submonoid.closure s = ⊤) {f g : M →* N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• AddMonoidHom.eqOn_closureM 📋 Mathlib.Algebra.Group.Submonoid.Basic
  {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] {f g : M →+ N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(AddSubmonoid.closure s)
• MonoidHom.eqOn_closureM 📋 Mathlib.Algebra.Group.Submonoid.Basic
  {M : Type u_1} {N : Type u_2} [MulOneClass M] [MulOneClass N] {f g : M →* N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Submonoid.closure s)
• AddMonoidHom.eq_of_eqOn_top 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {M : Type u_6} [AddMonoid M] {f g : G →+ M} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• MonoidHom.eq_of_eqOn_top 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [Group G] {M : Type u_6} [Monoid M] {f g : G →* M} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• AddMonoidHom.eq_of_eqOn_dense 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {M : Type u_6} [AddMonoid M] {s : Set G} (hs : AddSubgroup.closure s = ⊤) {f g : G →+ M} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• MonoidHom.eq_of_eqOn_dense 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [Group G] {M : Type u_6} [Monoid M] {s : Set G} (hs : Subgroup.closure s = ⊤) {f g : G →* M} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• AddMonoidHom.eqOn_closure 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {M : Type u_6} [AddMonoid M] {f g : G →+ M} {s : Set G} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(AddSubgroup.closure s)
• MonoidHom.eqOn_closure 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [Group G] {M : Type u_6} [Monoid M] {f g : G →* M} {s : Set G} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subgroup.closure s)
• NonUnitalRingHom.eq_of_eqOn_stop 📋 Mathlib.RingTheory.NonUnitalSubsemiring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] {f g : F} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• NonUnitalRingHom.eq_of_eqOn_sdense 📋 Mathlib.RingTheory.NonUnitalSubsemiring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {s : Set R} (hs : NonUnitalSubsemiring.closure s = ⊤) {f g : F} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• NonUnitalRingHom.eqOn_sclosure 📋 Mathlib.RingTheory.NonUnitalSubsemiring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {f g : F} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(NonUnitalSubsemiring.closure s)
• RingHom.eq_of_eqOn_stop 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f g : R →+* S} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• RingHom.eq_of_eqOn_sdense 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Set R} (hs : Subsemiring.closure s = ⊤) {f g : R →+* S} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• RingHom.eqOn_sclosure 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f g : R →+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subsemiring.closure s)
• NonUnitalRingHom.eq_of_eqOn_set_top 📋 Mathlib.RingTheory.NonUnitalSubring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f g : R →ₙ+* S} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• NonUnitalRingHom.eq_of_eqOn_set_dense 📋 Mathlib.RingTheory.NonUnitalSubring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {s : Set R} (hs : NonUnitalSubring.closure s = ⊤) {f g : R →ₙ+* S} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• NonUnitalRingHom.eqOn_set_closure 📋 Mathlib.RingTheory.NonUnitalSubring.Basic
  {R : Type u} {S : Type v} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] {f g : R →ₙ+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(NonUnitalSubring.closure s)
• RingHom.eq_of_eqOn_set_top 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} [NonAssocRing R] {S : Type v} [Semiring S] {f g : R →+* S} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• RingHom.eq_of_eqOn_set_dense 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} [NonAssocRing R] {S : Type v} [Semiring S] {s : Set R} (hs : Subring.closure s = ⊤) {f g : R →+* S} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• RingHom.eqOn_set_closure 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} [NonAssocRing R] {S : Type v} [Semiring S] {f g : R →+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subring.closure s)
• LinearMap.eqOn_eqLocus 📋 Mathlib.Algebra.Module.Submodule.EqLocus
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f g : M →ₛₗ[τ₁₂] M₂} : Set.EqOn ⇑f ⇑g ↑(f.eqLocus g)
• LinearMap.ext_on_codisjoint 📋 Mathlib.Algebra.Module.Submodule.EqLocus
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_5} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f g : F} {S T : Submodule R M} (hST : Codisjoint S T) (hS : Set.EqOn ⇑f ⇑g ↑S) (hT : Set.EqOn ⇑f ⇑g ↑T) : f = g
• LinearMap.le_eqLocus 📋 Mathlib.Algebra.Module.Submodule.EqLocus
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f g : M →ₛₗ[τ₁₂] M₂} {S : Submodule R M} : S ≤ f.eqLocus g ↔ Set.EqOn ⇑f ⇑g ↑S
• LinearMap.eqOn_sup 📋 Mathlib.Algebra.Module.Submodule.EqLocus
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {F : Type u_5} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂] {f g : F} {S T : Submodule R M} (hS : Set.EqOn ⇑f ⇑g ↑S) (hT : Set.EqOn ⇑f ⇑g ↑T) : Set.EqOn ⇑f ⇑g ↑(S ⊔ T)
• LinearMap.ext_on 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {s : Set M} {f g : M →ₛₗ[σ₁₂] M₂} (hv : Submodule.span R s = ⊤) (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• LinearMap.eqOn_span' 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {s : Set M} {f g : M →ₛₗ[σ₁₂] M₂} (H : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Submodule.span R s)
• LinearMap.eqOn_span_iff 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {s : Set M} {f g : M →ₛₗ[σ₁₂] M₂} : Set.EqOn ⇑f ⇑g ↑(Submodule.span R s) ↔ Set.EqOn (⇑f) (⇑g) s
• LinearMap.eqOn_span 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {s : Set M} {f g : M →ₛₗ[σ₁₂] M₂} (H : Set.EqOn (⇑f) (⇑g) s) ⦃x : M⦄ (h : x ∈ Submodule.span R s) : f x = g x
• Set.eqOn_indicator 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} : Set.EqOn (s.indicator f) f s
• Set.eqOn_mulIndicator 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} : Set.EqOn (s.mulIndicator f) f s
• Set.indicator_congr 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f g : α → M} (h : Set.EqOn f g s) : s.indicator f = s.indicator g
• Set.mulIndicator_congr 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f g : α → M} (h : Set.EqOn f g s) : s.mulIndicator f = s.mulIndicator g
• Set.eqOn_indicator' 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [Zero M] {s : Set α} {f : α → M} : Set.EqOn (s.indicator f) 0 sᶜ
• Set.eqOn_mulIndicator' 📋 Mathlib.Algebra.Notation.Indicator
  {α : Type u_1} {M : Type u_3} [One M] {s : Set α} {f : α → M} : Set.EqOn (s.mulIndicator f) 1 sᶜ
• eqOn_fun_finsetProd 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_8} {α : Type u_9} {β : Type u_10} [CommMonoid α] {s : Set β} {f f' : ι → β → α} (h : ∀ (i : ι), Set.EqOn (f i) (f' i) s) (v : Finset ι) : Set.EqOn (fun b => ∏ i ∈ v, f i b) (fun b => ∏ i ∈ v, f' i b) s
• eqOn_fun_finsetSum 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_8} {α : Type u_9} {β : Type u_10} [AddCommMonoid α] {s : Set β} {f f' : ι → β → α} (h : ∀ (i : ι), Set.EqOn (f i) (f' i) s) (v : Finset ι) : Set.EqOn (fun b => ∑ i ∈ v, f i b) (fun b => ∑ i ∈ v, f' i b) s
• eqOn_finsetProd 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_8} {α : Type u_9} {β : Type u_10} [CommMonoid α] {s : Set β} {f f' : ι → β → α} (h : ∀ (i : ι), Set.EqOn (f i) (f' i) s) (v : Finset ι) : Set.EqOn (∏ i ∈ v, f i) (∏ i ∈ v, f' i) s
• eqOn_finsetSum 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_8} {α : Type u_9} {β : Type u_10} [AddCommMonoid α] {s : Set β} {f f' : ι → β → α} (h : ∀ (i : ι), Set.EqOn (f i) (f' i) s) (v : Finset ι) : Set.EqOn (∑ i ∈ v, f i) (∑ i ∈ v, f' i) s
• finprod_mem_of_eqOn_one 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {M : Type u_5} [CommMonoid M] {f : α → M} {s : Set α} (hf : Set.EqOn f 1 s) : ∏ᶠ (i : α) (_ : i ∈ s), f i = 1
• finsum_mem_of_eqOn_zero 📋 Mathlib.Algebra.BigOperators.Finprod
  {α : Type u_1} {M : Type u_5} [AddCommMonoid M] {f : α → M} {s : Set α} (hf : Set.EqOn f 0 s) : ∑ᶠ (i : α) (_ : i ∈ s), f i = 0
• LinearIndepOn.congr 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
  {ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {w : ι → M} (hli : LinearIndepOn R v s) (h : Set.EqOn v w s) : LinearIndepOn R w s
• linearIndepOn_congr 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
  {ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {w : ι → M} (h : Set.EqOn v w s) : LinearIndepOn R v s ↔ LinearIndepOn R w s
• Set.EqOn.eventuallyEq 📋 Mathlib.Order.Filter.Basic
  {α : Type u_1} {β : Type u_2} {s : Set α} {f g : α → β} (h : Set.EqOn f g s) : f =ᶠ[Filter.principal s] g
• Filter.eventuallyEq_principal 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {s : Set α} {f g : α → β} : f =ᶠ[Filter.principal s] g ↔ Set.EqOn f g s
• Filter.eventuallyEq_of_mem 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} {s : Set α} (hs : s ∈ l) (h : Set.EqOn f g s) : f =ᶠ[l] g
• Set.EqOn.eventuallyEq_of_mem 📋 Mathlib.Order.Filter.Basic
  {α : Type u_1} {β : Type u_2} {s : Set α} {l : Filter α} {f g : α → β} (h : Set.EqOn f g s) (hl : s ∈ l) : f =ᶠ[l] g
• Filter.EventuallyEq.exists_mem 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) : ∃ s ∈ l, Set.EqOn f g s
• Filter.eventuallyEq_iff_exists_mem 📋 Mathlib.Order.Filter.Basic
  {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} : f =ᶠ[l] g ↔ ∃ s ∈ l, Set.EqOn f g s
• AlgHom.le_equalizer 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {φ ψ : A →ₐ[R] B} {S : Subalgebra R A} : S ≤ φ.equalizer ψ ↔ Set.EqOn ⇑φ ⇑ψ ↑S
• AlgHom.adjoin_le_equalizer 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ₁ φ₂ : A →ₐ[R] B) {s : Set A} (h : Set.EqOn (⇑φ₁) (⇑φ₂) s) : Algebra.adjoin R s ≤ φ₁.equalizer φ₂
• AlgHom.eqOn_adjoin_iff 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ ψ : A →ₐ[R] B} {s : Set A} : Set.EqOn ⇑φ ⇑ψ ↑(Algebra.adjoin R s) ↔ Set.EqOn (⇑φ) (⇑ψ) s
• AlgHom.ext_on_codisjoint 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {F : Type u_4} [FunLike F A B] [AlgHomClass F R A B] {φ ψ : F} {S T : Subalgebra R A} (hST : Codisjoint S T) (hS : Set.EqOn ⇑φ ⇑ψ ↑S) (hT : Set.EqOn ⇑φ ⇑ψ ↑T) : φ = ψ
• AlgHom.eqOn_sup 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {F : Type u_4} [FunLike F A B] [AlgHomClass F R A B] {φ ψ : F} {S T : Subalgebra R A} (hS : Set.EqOn ⇑φ ⇑ψ ↑S) (hT : Set.EqOn ⇑φ ⇑ψ ↑T) : Set.EqOn ⇑φ ⇑ψ ↑(S ⊔ T)
• AlgHom.ext_of_adjoin_eq_top 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {s : Set A} (h : Algebra.adjoin R s = ⊤) ⦃φ₁ φ₂ : A →ₐ[R] B⦄ (hs : Set.EqOn (⇑φ₁) (⇑φ₂) s) : φ₁ = φ₂
• RingHom.eq_of_eqOn_subfield_top 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {f g : K →+* L} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g
• RingHom.eq_of_eqOn_of_field_closure_eq_top 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {s : Set K} (hs : Subfield.closure s = ⊤) {f g : K →+* L} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• RingHom.eqOn_field_closure 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} [DivisionRing K] {L : Type v} [Semiring L] {f g : K →+* L} {s : Set K} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subfield.closure s)
• StarAlgHom.adjoin_le_equalizer 📋 Mathlib.Algebra.Star.Subalgebra
  {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R A] [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] (f g : F) {s : Set A} (h : Set.EqOn (⇑f) (⇑g) s) : StarAlgebra.adjoin R s ≤ StarAlgHom.equalizer f g
• StarAlgHom.ext_of_adjoin_eq_top 📋 Mathlib.Algebra.Star.Subalgebra
  {F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [Semiring B] [Algebra R B] [StarRing B] [StarModule R A] [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] {s : Set A} (h : StarAlgebra.adjoin R s = ⊤) ⦃f g : F⦄ (hs : Set.EqOn (⇑f) (⇑g) s) : f = g
• Filter.EventuallyEq.self_of_nhdsSet 📋 Mathlib.Topology.NhdsSet
  {X : Type u_2} [TopologicalSpace X] {s : Set X} {Y : Type u_4} {f g : X → Y} (h : f =ᶠ[nhdsSet s] g) : Set.EqOn f g s
• eventuallyEq_nhdsWithin_of_eqOn 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α} (h : Set.EqOn f g s) : f =ᶠ[nhdsWithin a s] g
• Set.EqOn.eventuallyEq_nhdsWithin 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {f g : α → β} {s : Set α} {a : α} (h : Set.EqOn f g s) : f =ᶠ[nhdsWithin a s] g
• ContinuousOn.congr 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : Set.EqOn g f s) : ContinuousOn g s
• continuousOn_congr 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s : Set α} (h' : Set.EqOn g f s) : ContinuousOn g s ↔ ContinuousOn f s
• Continuous.tendsto_nhdsSet_nhds 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {b : β} {f : α → β} (h : Continuous f) (h' : Set.EqOn f (fun x => b) s) : Filter.Tendsto f (nhdsSet s) (nhds b)
• ContinuousOn.congr_mono 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s) (h' : Set.EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁
• ContinuousWithinAt.congr_mono 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s₁ : Set α} {x : α} (h : ContinuousWithinAt f s x) (h' : Set.EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) : ContinuousWithinAt g s₁ x
• continuousOn_piecewise_ite 📋 Mathlib.Topology.Piecewise
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s' t : Set α} [(x : α) → Decidable (x ∈ t)] (h : ContinuousOn f s) (h' : ContinuousOn g s') (H : s ∩ frontier t = s' ∩ frontier t) (Heq : Set.EqOn f g (s ∩ frontier t)) : ContinuousOn (t.piecewise f g) (t.ite s s')
• continuousOn_piecewise_ite' 📋 Mathlib.Topology.Piecewise
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : α → β} {s s' t : Set α} [(x : α) → Decidable (x ∈ t)] (h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn g (s' ∩ closure tᶜ)) (H : s ∩ frontier t = s' ∩ frontier t) (Heq : Set.EqOn f g (s ∩ frontier t)) : ContinuousOn (t.piecewise f g) (t.ite s s')
• ContinuousWithinAt.eqOn_const_closure 📋 Mathlib.Topology.Separation.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {f : X → Y} {s : Set X} {c : Y} (h : ∀ x ∈ closure s, ContinuousWithinAt f s x) (ht : Set.EqOn f (fun x => c) s) : Set.EqOn f (fun x => c) (closure s)
• Continuous.ext_on 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : Set.EqOn f g s) : f = g
• Set.EqOn.closure 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] {s : Set Y} {f g : Y → X} (h : Set.EqOn f g s) (hf : Continuous f) (hg : Continuous g) : Set.EqOn f g (closure s)
• Set.EqOn.of_subset_closure 📋 Mathlib.Topology.Separation.Hausdorff
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y] {s t : Set X} {f g : X → Y} (h : Set.EqOn f g s) (hf : ContinuousOn f t) (hg : ContinuousOn g t) (hst : s ⊆ t) (hts : t ⊆ closure s) : Set.EqOn f g t
• IsPreconnected.eqOn_const_of_mapsTo 📋 Mathlib.Topology.Connected.TotallyDisconnected
  {α : Type u} [TopologicalSpace α] {S : Set α} (hS : IsPreconnected S) {β : Type u_3} [TopologicalSpace β] {T : Set β} (hT : IsDiscrete T) {f : α → β} (hc : ContinuousOn f S) (hTm : Set.MapsTo f S T) (hne : T.Nonempty) : ∃ y ∈ T, Set.EqOn f (Function.const α y) S
• UniformContinuousOn.congr 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [UniformSpace β] {f g : α → β} {s : Set α} (hf : UniformContinuousOn f s) (h : Set.EqOn f g s) : UniformContinuousOn g s
• Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq 📋 Mathlib.Topology.UniformSpace.Basic
  {α : Type ua} {β : Type ub} [UniformSpace α] [TopologicalSpace β] {r : SetRel α α} {s : Set β} {f g : β → α} (hf : ∀ x ∈ s, ContinuousAt f x) (hg : ∀ x ∈ s, ContinuousAt g x) (hfg : Set.EqOn f g s) (hr : r ∈ uniformity α) : ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, (f x, g x) ∈ r
• TendstoUniformlyOn.congr_right 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} {g : α → β} (hf : TendstoUniformlyOn F f p s) (hfg : Set.EqOn f g s) : TendstoUniformlyOn F g p s
• TendstoUniformlyOn.congr 📋 Mathlib.Topology.UniformSpace.UniformConvergence
  {α : Type u_1} {β : Type u_2} {ι : Type u_4} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s) (hff' : ∀ᶠ (n : ι) in p, Set.EqOn (F n) (F' n) s) : TendstoUniformlyOn F' f p s
• TendstoLocallyUniformlyOn.congr_right 📋 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} {s : Set α} {p : Filter ι} {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : Set.EqOn f g s) : TendstoLocallyUniformlyOn F g p s
• TendstoLocallyUniformlyOn.congr 📋 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} {s : Set α} {p : Filter ι} {G : ι → α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : ∀ (n : ι), Set.EqOn (F n) (G n) s) : TendstoLocallyUniformlyOn G f p s
• TendstoLocallyUniformlyOn.unique 📋 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} {s : Set α} {p : Filter ι} [p.NeBot] [T2Space β] {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s) (hg : TendstoLocallyUniformlyOn F g p s) : Set.EqOn f g s
• IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq 📋 Mathlib.Topology.Algebra.Field
  {α : Type u_2} {𝕜 : Type u_3} {f : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Ring 𝕜] [NoZeroDivisors 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hsq : Set.EqOn (f ^ 2) 1 S) : Set.EqOn f 1 S ∨ Set.EqOn f (-1) S
• IsPreconnected.eq_of_sq_eq 📋 Mathlib.Topology.Algebra.Field
  {α : Type u_2} {𝕜 : Type u_3} {f g : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Field 𝕜] [ContinuousInv₀ 𝕜] [ContinuousMul 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) {y : α} (hy : y ∈ S) (hy' : f y = g y) : Set.EqOn f g S
• IsPreconnected.eq_or_eq_neg_of_sq_eq 📋 Mathlib.Topology.Algebra.Field
  {α : Type u_2} {𝕜 : Type u_3} {f g : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Field 𝕜] [ContinuousInv₀ 𝕜] [ContinuousMul 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) : Set.EqOn f g S ∨ Set.EqOn f (-g) S
• PartialEquiv.EqOnSource.eqOn 📋 Mathlib.Logic.Equiv.PartialEquiv
  {α : Type u_1} {β : Type u_2} {e e' : PartialEquiv α β} (h : e ≈ e') : Set.EqOn (↑e) (↑e') e.source
• PartialEquiv.EqOnSource.symm_eqOn 📋 Mathlib.Logic.Equiv.PartialEquiv
  {α : Type u_1} {β : Type u_2} {e e' : PartialEquiv α β} (h : e ≈ e') : Set.EqOn (↑e.symm) (↑e'.symm) e.target
• PartialEquiv.IsImage.symm_eq_on_of_inter_eq_of_eqOn 📋 Mathlib.Logic.Equiv.PartialEquiv
  {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {e' : PartialEquiv α β} (h : e.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : Set.EqOn (↑e.symm) (↑e'.symm) (e.target ∩ t)
• PartialEquiv.IsImage.inter_eq_of_inter_eq_of_eqOn 📋 Mathlib.Logic.Equiv.PartialEquiv
  {α : Type u_1} {β : Type u_2} {e : PartialEquiv α β} {s : Set α} {t : Set β} {e' : PartialEquiv α β} (h : e.IsImage s t) (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : e.target ∩ t = e'.target ∩ t
• Derivation.eqOn_adjoin 📋 Mathlib.RingTheory.Derivation.Basic
  {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} {s : Set A} (h : Set.EqOn (⇑D1) (⇑D2) s) : Set.EqOn ⇑D1 ⇑D2 ↑(Algebra.adjoin R s)
• Derivation.ext_of_adjoin_eq_top 📋 Mathlib.RingTheory.Derivation.Basic
  {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M] {D1 D2 : Derivation R A M} (s : Set A) (hs : Algebra.adjoin R s = ⊤) (h : Set.EqOn (⇑D1) (⇑D2) s) : D1 = D2
• ContinuousLinearMap.ext_on 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} (hs : Dense ↑(Submodule.span R₁ s)) {f g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• ContinuousLinearMap.eqOn_closure_span 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} {f g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn (⇑f) (⇑g) (closure ↑(Submodule.span R₁ s))
• Equidecomp.IsDecompOn.mono 📋 Mathlib.Algebra.Group.Action.Equidecomp
  {X : Type u_1} {G : Type u_2} [SMul G X] {f f' : X → X} {A A' : Set X} {S : Finset G} (h : Equidecomp.IsDecompOn f A S) (hA' : A' ⊆ A) (hf' : Set.EqOn f f' A') : Equidecomp.IsDecompOn f' A' S
• LieDerivation.ext_of_lieSpan_eq_top 📋 Mathlib.Algebra.Lie.Derivation.Basic
  {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} (s : Set L) (hs : LieSubalgebra.lieSpan R L s = ⊤) (h : Set.EqOn (⇑D1) (⇑D2) s) : D1 = D2
• LieDerivation.eqOn_lieSpan 📋 Mathlib.Algebra.Lie.Derivation.Basic
  {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {D1 D2 : LieDerivation R L M} {s : Set L} (h : Set.EqOn (⇑D1) (⇑D2) s) : Set.EqOn ⇑D1 ⇑D2 ↑(LieSubalgebra.lieSpan R L s)
• LinearRecurrence.eq_iff_eqOn_range_order 📋 Mathlib.Algebra.LinearRecurrence
  {R : Type u_1} [CommSemiring R] (E : LinearRecurrence R) (u v : ℕ → R) (hu : E.IsSolution u) (hv : E.IsSolution v) : u = v ↔ Set.EqOn u v ↑(Finset.range E.order)
• LinearRecurrence.sol_eq_of_eq_init 📋 Mathlib.Algebra.LinearRecurrence
  {R : Type u_1} [CommSemiring R] (E : LinearRecurrence R) (u v : ℕ → R) (hu : E.IsSolution u) (hv : E.IsSolution v) : u = v ↔ Set.EqOn u v ↑(Finset.range E.order)
• LinearMap.FiniteRangeSetoid.equiv_of_eqOn_coFG 📋 Mathlib.Algebra.Module.LinearMap.FiniteRange
  {K : Type u_1} {V : Type u_2} {V₂ : Type u_3} [CommRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₂] [Module K V₂] [IsNoetherianRing K] {u v : V →ₗ[K] V₂} {A : Submodule K V} (A_coFG : A.CoFG) (eqOn_A : Set.EqOn ⇑u ⇑v ↑A) : u ≈ v
• LinearMap.FiniteRangeSetoid.equiv_of_eqOn_of_isNoetherian 📋 Mathlib.Algebra.Module.LinearMap.FiniteRange
  {K : Type u_1} {V : Type u_2} {V₂ : Type u_3} [CommRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₂] [Module K V₂] {u v : V →ₗ[K] V₂} (A : Submodule K V) [quot_A_noeth : IsNoetherian K (V ⧸ A)] (eqOn_A : Set.EqOn ⇑u ⇑v ↑A) : u ≈ v
• MvPolynomial.derivation_eqOn_supported 📋 Mathlib.Algebra.MvPolynomial.Derivation
  {σ : Type u_1} {R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Module (MvPolynomial σ R) A] {D₁ D₂ : Derivation R (MvPolynomial σ R) A} {s : Set σ} (h : Set.EqOn (⇑D₁ ∘ MvPolynomial.X) (⇑D₂ ∘ MvPolynomial.X) s) {f : MvPolynomial σ R} (hf : f ∈ MvPolynomial.supported R s) : D₁ f = D₂ f
• LipschitzOnWith.extend_real 📋 Mathlib.Topology.MetricSpace.Lipschitz
  {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} {s : Set α} {K : NNReal} (hf : LipschitzOnWith K f s) : ∃ g, LipschitzWith K g ∧ Set.EqOn f g s
• LipschitzOnWith.extend_pi 📋 Mathlib.Topology.MetricSpace.Lipschitz
  {α : Type u} {ι : Type x} [PseudoMetricSpace α] [Fintype ι] {f : α → ι → ℝ} {s : Set α} {K : NNReal} (hf : LipschitzOnWith K f s) : ∃ g, LipschitzWith K g ∧ Set.EqOn f g s
• exists_measurable_piecewise 📋 Mathlib.MeasureTheory.MeasurableSpace.Constructions
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {ι : Type u_6} [Countable ι] [Nonempty ι] (t : ι → Set α) (t_meas : ∀ (n : ι), MeasurableSet (t n)) (g : ι → α → β) (hg : ∀ (n : ι), Measurable (g n)) (ht : Pairwise fun i j => Set.EqOn (g i) (g j) (t i ∩ t j)) : ∃ f, Measurable f ∧ ∀ (n : ι), Set.EqOn f (g n) (t n)
• ContinuousAlgHom.ext_on 📋 Mathlib.Topology.Algebra.Algebra
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [T2Space B] {s : Set A} (hs : Dense ↑(Algebra.adjoin R s)) {f g : A →A[R] B} (h : Set.EqOn (⇑f) (⇑g) s) : f = g
• ContinuousAlgHom.eqOn_closure_adjoin 📋 Mathlib.Topology.Algebra.Algebra
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [T2Space B] {s : Set A} {f g : A →A[R] B} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn (⇑f) (⇑g) (closure ↑(Algebra.adjoin R s))
• Set.EqOn.aeEq 📋 Mathlib.MeasureTheory.OuterMeasure.AE
  {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F (Set α) ENNReal] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α} {f g : α → β} (h : Set.EqOn f g s) (h2 : μ sᶜ = 0) : f =ᵐ[μ] g
• Set.EqOn.aeEq_restrict 📋 Mathlib.MeasureTheory.Measure.Restrict
  {α : Type u_6} {β : Type u_7} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → β} (h : Set.EqOn f g s) (hs : MeasurableSet s) : f =ᵐ[μ.restrict s] g
• Measurable.isGLB_of_mem 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Order
  {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsGLB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g') : Measurable g
• Measurable.isLUB_of_mem 📋 Mathlib.MeasureTheory.Constructions.BorelSpace.Order
  {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g') : Measurable g
• MeasureTheory.setLIntegral_congr_fun 📋 Mathlib.MeasureTheory.Integral.Lebesgue.Basic
  {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ENNReal} {s : Set α} (hs : MeasurableSet s) (hfg : Set.EqOn f g s) : ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
• MeasureTheory.setLIntegral_eq_zero 📋 Mathlib.MeasureTheory.Integral.Lebesgue.Basic
  {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {s : Set α} (hs : MeasurableSet s) (h's : Set.EqOn f 0 s) : ∫⁻ (x : α) in s, f x ∂μ = 0
• MeasureTheory.Measure.eqOn_open_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {U : Set X} {f g : X → Y} (h : f =ᵐ[μ.restrict U] g) (hU : IsOpen U) (hf : ContinuousOn f U) (hg : ContinuousOn g U) : Set.EqOn f g U
• MeasureTheory.Measure.eqOn_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure] {s : Set X} {f g : X → Y} (h : f =ᵐ[μ.restrict s] g) (hf : ContinuousOn f s) (hg : ContinuousOn g s) (hU : s ⊆ closure (interior s)) : Set.EqOn f g s
• MeasureTheory.Measure.eqOn_Ioo_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ioo a b)] g) (hf : ContinuousOn f (Set.Ioo a b)) (hg : ContinuousOn g (Set.Ioo a b)) : Set.EqOn f g (Set.Ioo a b)
• MeasureTheory.Measure.eqOn_Ico_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ico a b)] g) (hf : ContinuousOn f (Set.Ico a b)) (hg : ContinuousOn g (Set.Ico a b)) : Set.EqOn f g (Set.Ico a b)
• MeasureTheory.Measure.eqOn_Ioc_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Ioc a b)] g) (hf : ContinuousOn f (Set.Ioc a b)) (hg : ContinuousOn g (Set.Ioc a b)) : Set.EqOn f g (Set.Ioc a b)
• MeasureTheory.Measure.eqOn_Icc_of_ae_eq 📋 Mathlib.MeasureTheory.Measure.OpenPos
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] {m : MeasurableSpace X} [TopologicalSpace Y] [T2Space Y] (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure] [DenselyOrdered X] {a b : X} (hne : a ≠ b) {f g : X → Y} (hfg : f =ᵐ[μ.restrict (Set.Icc a b)] g) (hf : ContinuousOn f (Set.Icc a b)) (hg : ContinuousOn g (Set.Icc a b)) : Set.EqOn f g (Set.Icc a b)
• Complex.norm_sub_one_sq_eqOn_sphere 📋 Mathlib.Analysis.Complex.Norm
  : Set.EqOn (fun x => ‖x - 1‖ ^ 2) (fun z => 2 * (1 - z.re)) (Metric.sphere 0 1)
• ConcaveOn.congr 📋 Mathlib.Analysis.Convex.Function
  {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [AddCommMonoid β] [PartialOrder β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s) : ConcaveOn 𝕜 s g
• ConvexOn.congr 📋 Mathlib.Analysis.Convex.Function
  {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [AddCommMonoid β] [PartialOrder β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConvexOn 𝕜 s f) (hfg : Set.EqOn f g s) : ConvexOn 𝕜 s g
• StrictConcaveOn.congr 📋 Mathlib.Analysis.Convex.Function
  {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [AddCommMonoid β] [PartialOrder β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s) : StrictConcaveOn 𝕜 s g
• StrictConvexOn.congr 📋 Mathlib.Analysis.Convex.Function
  {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [AddCommMonoid β] [PartialOrder β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConvexOn 𝕜 s f) (hfg : Set.EqOn f g s) : StrictConvexOn 𝕜 s g
• AffineMap.eqOn_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {V₂ : Type u_8} {P₂ : Type u_9} [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {s : Set P₁} {f g : P₁ →ᵃ[k] P₂} (h_agree : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(affineSpan k s)
• AffineMap.linear_eqOn_vectorSpan 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
  {k : Type u_1} {V₁ : Type u_2} {P₁ : Type u_3} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] {V₂ : Type u_8} {P₂ : Type u_9} [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] {s : Set P₁} {f g : P₁ →ᵃ[k] P₂} (h_agree : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f.linear ⇑g.linear ↑(vectorSpan k s)

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