ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UniformOnFun.continuousOn_eval₂
protected theorem continuousOn_eval₂ [TopologicalSpace α] (h𝔖 : ∀ x, ∃ V ∈ 𝔖, V ∈ 𝓝 x) :
    ContinuousOn (fun fx : (α →ᵤ[𝔖] β) × α ↦ toFun 𝔖 fx.1 fx.2)
      {fx | ContinuousAt (toFun 𝔖 fx.1) fx.2} 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [TopologicalSpace α] (h𝔖 : ∀ (x : α), ∃ V ∈ 𝔖, V ∈ 𝓝 x)
Goal: ContinuousOn (fun fx => (UniformOnFun.toFun 𝔖) fx.1 fx.2) {fx | ContinuousAt ((UniformOnFun.toFun 𝔖) fx.1) fx.2}
-- UniformOnFun.tendsto_iff_tendstoUniformlyOn
protected theorem tendsto_iff_tendstoUniformlyOn {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
    Tendsto F p (𝓝 f) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (toFun 𝔖 ∘ F) (toFun 𝔖 f) p s 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} {p : Filter ι} [UniformSpace β] {𝔖 : Set (Set α)} {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β}
Goal: Filter.Tendsto F p (𝓝 f) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (⇑(UniformOnFun.toFun 𝔖) ∘ F) ((UniformOnFun.toFun 𝔖) f) p s
-- UniformOnFun.continuous_rng_iff
protected lemma continuous_rng_iff {X : Type*} [TopologicalSpace X] {f : X → (α →ᵤ[𝔖] β)} :
    Continuous f ↔ ∀ s ∈ 𝔖,
      Continuous (UniformFun.ofFun ∘ s.restrict ∘ UniformOnFun.toFun 𝔖 ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} {X : Type u_5} [TopologicalSpace X] {f : X → α →ᵤ[𝔖] β}
Goal: Continuous f ↔ ∀ s ∈ 𝔖, Continuous (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(UniformOnFun.toFun 𝔖) ∘ f)
-- UniformOnFun.isClosed_setOf_continuous
theorem isClosed_setOf_continuous [TopologicalSpace α] (h : RestrictGenTopology 𝔖) :
    IsClosed {f : α →ᵤ[𝔖] β | Continuous (toFun 𝔖 f)} 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [TopologicalSpace α] (h : RestrictGenTopology 𝔖)
Goal: IsClosed {f | Continuous ((UniformOnFun.toFun 𝔖) f)}
-- UniformOnFun.isClosed_setOf_continuous_of_le
@[deprecated isClosed_setOf_continuous (since := "2024-06-29")]
theorem isClosed_setOf_continuous_of_le [t : TopologicalSpace α]
    (h : t ≤ ⨆ s ∈ 𝔖, .coinduced (Subtype.val : s → α) inferInstance) :
    IsClosed {f : α →ᵤ[𝔖] β | Continuous (toFun 𝔖 f)} 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace β] {𝔖 : Set (Set α)} [t : TopologicalSpace α] (h : t ≤ ⨆ s ∈ 𝔖, TopologicalSpace.coinduced Subtype.val inferInstance)
Goal: IsClosed {f | Continuous ((UniformOnFun.toFun 𝔖) f)}
-- UniformOnFun.uniformSpace_eq_inf_precomp_of_cover
theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
    (𝔗₁ : Set (Set δ₁)) (𝔗₂ : Set (Set δ₂))
    (h_image₁ : MapsTo (φ₁ '' ·) 𝔗₁ 𝔖) (h_image₂ : MapsTo (φ₂ '' ·) 𝔗₂ 𝔖)
    (h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁) (h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂)
    (h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂) :
    𝒱(α, β, 𝔖, _) =
      .comap (ofFun 𝔗₁ ∘ (· ∘ φ₁) ∘ toFun 𝔖) 𝒱(δ₁, β, 𝔗₁, _) ⊓
      .comap (ofFun 𝔗₂ ∘ (· ∘ φ₂) ∘ toFun 𝔖) 𝒱(δ₂, β, 𝔗₂, _) 
Arguments: {α : Type u_1} {β : Type u_2} [UniformSpace β] (𝔖 : Set (Set α)) {δ₁ : Type u_5} {δ₂ : Type u_6} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α) (𝔗₁ : Set (Set δ₁)) (𝔗₂ : Set (Set δ₂)) (h_image₁ : Set.MapsTo (fun x => φ₁ '' x) 𝔗₁ 𝔖) (h_image₂ : Set.MapsTo (fun x => φ₂ '' x) 𝔗₂ 𝔖) (h_preimage₁ : Set.MapsTo (fun x => φ₁ ⁻¹' x) 𝔖 𝔗₁) (h_preimage₂ : Set.MapsTo (fun x => φ₂ ⁻¹' x) 𝔖 𝔗₂) (h_cover : ∀ S ∈ 𝔖, S ⊆ Set.range φ₁ ∪ Set.range φ₂)
Goal: UniformOnFun.uniformSpace α β 𝔖 =
  UniformSpace.comap (⇑(UniformOnFun.ofFun 𝔗₁) ∘ (fun x => x ∘ φ₁) ∘ ⇑(UniformOnFun.toFun 𝔖))
      (UniformOnFun.uniformSpace δ₁ β 𝔗₁) ⊓
    UniformSpace.comap (⇑(UniformOnFun.ofFun 𝔗₂) ∘ (fun x => x ∘ φ₂) ∘ ⇑(UniformOnFun.toFun 𝔖))
      (UniformOnFun.uniformSpace δ₂ β 𝔗₂)
-- UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover
theorem uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} (φ : Π i, δ i → α)
    (𝔗 : ∀ i, Set (Set (δ i))) (h_image : ∀ i, MapsTo (φ i '' ·) (𝔗 i) 𝔖)
    (h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i))
    (h_cover : ∀ S ∈ 𝔖, ∃ I : Set ι, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i)) :
    𝒱(α, β, 𝔖, _) = ⨅ i, .comap (ofFun (𝔗 i) ∘ (· ∘ φ i) ∘ toFun 𝔖) 𝒱(δ i, β, 𝔗 i, _) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [UniformSpace β] (𝔖 : Set (Set α)) {δ : ι → Type u_5} (φ : (i : ι) → δ i → α) (𝔗 : (i : ι) → Set (Set (δ i))) (h_image : ∀ (i : ι), Set.MapsTo (fun x => φ i '' x) (𝔗 i) 𝔖) (h_preimage : ∀ (i : ι), Set.MapsTo (fun x => φ i ⁻¹' x) 𝔖 (𝔗 i)) (h_cover : ∀ S ∈ 𝔖, ∃ I, I.Finite ∧ S ⊆ ⋃ i ∈ I, Set.range (φ i))
Goal: UniformOnFun.uniformSpace α β 𝔖 =
  ⨅ i,
    UniformSpace.comap (⇑(UniformOnFun.ofFun (𝔗 i)) ∘ (fun x => x ∘ φ i) ∘ ⇑(UniformOnFun.toFun 𝔖))
      (UniformOnFun.uniformSpace (δ i) β (𝔗 i))
-- EquicontinuousAt.equicontinuousWithinAt
lemma EquicontinuousAt.equicontinuousWithinAt {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀)
    (S : Set X) : EquicontinuousWithinAt F S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀) (S : Set X)
Goal: EquicontinuousWithinAt F S x₀
-- EquicontinuousWithinAt.mono
lemma EquicontinuousWithinAt.mono {F : ι → X → α} {x₀ : X} {S T : Set X}
    (H : EquicontinuousWithinAt F T x₀) (hST : S ⊆ T) : EquicontinuousWithinAt F S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} {S T : Set X} (H : EquicontinuousWithinAt F T x₀) (hST : S ⊆ T)
Goal: EquicontinuousWithinAt F S x₀
-- equicontinuousWithinAt_univ
@[simp] lemma equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) :
    EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] (F : ι → X → α) (x₀ : X)
Goal: EquicontinuousWithinAt F Set.univ x₀ ↔ EquicontinuousAt F x₀
-- equicontinuousAt_restrict_iff
lemma equicontinuousAt_restrict_iff (F : ι → X → α) {S : Set X} (x₀ : S) :
    EquicontinuousAt (S.restrict ∘ F) x₀ ↔ EquicontinuousWithinAt F S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] (F : ι → X → α) {S : Set X} (x₀ : ↑S)
Goal: EquicontinuousAt (S.restrict ∘ F) x₀ ↔ EquicontinuousWithinAt F S ↑x₀
-- Equicontinuous.equicontinuousOn
lemma Equicontinuous.equicontinuousOn {F : ι → X → α} (H : Equicontinuous F)
    (S : Set X) : EquicontinuousOn F S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} (H : Equicontinuous F) (S : Set X)
Goal: EquicontinuousOn F S
-- EquicontinuousOn.mono
lemma EquicontinuousOn.mono {F : ι → X → α} {S T : Set X}
    (H : EquicontinuousOn F T) (hST : S ⊆ T) : EquicontinuousOn F S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S T : Set X} (H : EquicontinuousOn F T) (hST : S ⊆ T)
Goal: EquicontinuousOn F S
-- equicontinuousOn_univ
lemma equicontinuousOn_univ (F : ι → X → α) :
    EquicontinuousOn F univ ↔ Equicontinuous F 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] (F : ι → X → α)
Goal: EquicontinuousOn F Set.univ ↔ Equicontinuous F
-- equicontinuous_restrict_iff
lemma equicontinuous_restrict_iff (F : ι → X → α) {S : Set X} :
    Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] (F : ι → X → α) {S : Set X}
Goal: Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S
-- UniformEquicontinuous.uniformEquicontinuousOn
lemma UniformEquicontinuous.uniformEquicontinuousOn {F : ι → β → α} (H : UniformEquicontinuous F)
    (S : Set β) : UniformEquicontinuousOn F S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} (H : UniformEquicontinuous F) (S : Set β)
Goal: UniformEquicontinuousOn F S
-- UniformEquicontinuousOn.mono
lemma UniformEquicontinuousOn.mono {F : ι → β → α} {S T : Set β}
    (H : UniformEquicontinuousOn F T) (hST : S ⊆ T) : UniformEquicontinuousOn F S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S T : Set β} (H : UniformEquicontinuousOn F T) (hST : S ⊆ T)
Goal: UniformEquicontinuousOn F S
-- uniformEquicontinuousOn_univ
lemma uniformEquicontinuousOn_univ (F : ι → β → α) :
    UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] (F : ι → β → α)
Goal: UniformEquicontinuousOn F Set.univ ↔ UniformEquicontinuous F
-- uniformEquicontinuous_restrict_iff
lemma uniformEquicontinuous_restrict_iff (F : ι → β → α) {S : Set β} :
    UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] (F : ι → β → α) {S : Set β}
Goal: UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S
-- equicontinuousAt_empty
@[simp]
lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
    EquicontinuousAt F x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [h : IsEmpty ι] (F : ι → X → α) (x₀ : X)
Goal: EquicontinuousAt F x₀
-- equicontinuousWithinAt_empty
@[simp]
lemma equicontinuousWithinAt_empty [h : IsEmpty ι] (F : ι → X → α) (S : Set X) (x₀ : X) :
    EquicontinuousWithinAt F S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [h : IsEmpty ι] (F : ι → X → α) (S : Set X) (x₀ : X)
Goal: EquicontinuousWithinAt F S x₀
-- equicontinuous_empty
@[simp]
lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
    Equicontinuous F 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [IsEmpty ι] (F : ι → X → α)
Goal: Equicontinuous F
-- equicontinuousOn_empty
@[simp]
lemma equicontinuousOn_empty [IsEmpty ι] (F : ι → X → α) (S : Set X) :
    EquicontinuousOn F S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [IsEmpty ι] (F : ι → X → α) (S : Set X)
Goal: EquicontinuousOn F S
-- uniformEquicontinuous_empty
@[simp]
lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
    UniformEquicontinuous F 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [h : IsEmpty ι] (F : ι → β → α)
Goal: UniformEquicontinuous F
-- uniformEquicontinuousOn_empty
@[simp]
lemma uniformEquicontinuousOn_empty [h : IsEmpty ι] (F : ι → β → α) (S : Set β) :
    UniformEquicontinuousOn F S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [h : IsEmpty ι] (F : ι → β → α) (S : Set β)
Goal: UniformEquicontinuousOn F S
-- equicontinuousAt_finite
theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
    EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Finite ι] {F : ι → X → α} {x₀ : X}
Goal: EquicontinuousAt F x₀ ↔ ∀ (i : ι), ContinuousAt (F i) x₀
-- equicontinuousWithinAt_finite
theorem equicontinuousWithinAt_finite [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X} :
    EquicontinuousWithinAt F S x₀ ↔ ∀ i, ContinuousWithinAt (F i) S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X}
Goal: EquicontinuousWithinAt F S x₀ ↔ ∀ (i : ι), ContinuousWithinAt (F i) S x₀
-- equicontinuous_finite
theorem equicontinuous_finite [Finite ι] {F : ι → X → α} :
    Equicontinuous F ↔ ∀ i, Continuous (F i) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Finite ι] {F : ι → X → α}
Goal: Equicontinuous F ↔ ∀ (i : ι), Continuous (F i)
-- equicontinuousOn_finite
theorem equicontinuousOn_finite [Finite ι] {F : ι → X → α} {S : Set X} :
    EquicontinuousOn F S ↔ ∀ i, ContinuousOn (F i) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Finite ι] {F : ι → X → α} {S : Set X}
Goal: EquicontinuousOn F S ↔ ∀ (i : ι), ContinuousOn (F i) S
-- uniformEquicontinuous_finite
theorem uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
    UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [Finite ι] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ ∀ (i : ι), UniformContinuous (F i)
-- uniformEquicontinuousOn_finite
theorem uniformEquicontinuousOn_finite [Finite ι] {F : ι → β → α} {S : Set β} :
    UniformEquicontinuousOn F S ↔ ∀ i, UniformContinuousOn (F i) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [Finite ι] {F : ι → β → α} {S : Set β}
Goal: UniformEquicontinuousOn F S ↔ ∀ (i : ι), UniformContinuousOn (F i) S
-- equicontinuousAt_unique
theorem equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} :
    EquicontinuousAt F x ↔ ContinuousAt (F default) x 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Unique ι] {F : ι → X → α} {x : X}
Goal: EquicontinuousAt F x ↔ ContinuousAt (F default) x
-- equicontinuousWithinAt_unique
theorem equicontinuousWithinAt_unique [Unique ι] {F : ι → X → α} {S : Set X} {x : X} :
    EquicontinuousWithinAt F S x ↔ ContinuousWithinAt (F default) S x 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Unique ι] {F : ι → X → α} {S : Set X} {x : X}
Goal: EquicontinuousWithinAt F S x ↔ ContinuousWithinAt (F default) S x
-- equicontinuous_unique
theorem equicontinuous_unique [Unique ι] {F : ι → X → α} :
    Equicontinuous F ↔ Continuous (F default) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Unique ι] {F : ι → X → α}
Goal: Equicontinuous F ↔ Continuous (F default)
-- equicontinuousOn_unique
theorem equicontinuousOn_unique [Unique ι] {F : ι → X → α} {S : Set X} :
    EquicontinuousOn F S ↔ ContinuousOn (F default) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] [Unique ι] {F : ι → X → α} {S : Set X}
Goal: EquicontinuousOn F S ↔ ContinuousOn (F default) S
-- uniformEquicontinuous_unique
theorem uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} :
    UniformEquicontinuous F ↔ UniformContinuous (F default) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [Unique ι] {F : ι → β → α}
Goal: UniformEquicontinuous F ↔ UniformContinuous (F default)
-- uniformEquicontinuousOn_unique
theorem uniformEquicontinuousOn_unique [Unique ι] {F : ι → β → α} {S : Set β} :
    UniformEquicontinuousOn F S ↔ UniformContinuousOn (F default) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] [Unique ι] {F : ι → β → α} {S : Set β}
Goal: UniformEquicontinuousOn F S ↔ UniformContinuousOn (F default) S
-- equicontinuousWithinAt_iff_pair
theorem equicontinuousWithinAt_iff_pair {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S) :
    EquicontinuousWithinAt F S x₀ ↔
      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S)
Goal: EquicontinuousWithinAt F S x₀ ↔
  ∀ U ∈ uniformity α, ∃ V ∈ nhdsWithin x₀ S, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
-- equicontinuousAt_iff_pair
theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
    EquicontinuousAt F x₀ ↔
      ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X}
Goal: EquicontinuousAt F x₀ ↔ ∀ U ∈ uniformity α, ∃ V ∈ nhds x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U
-- UniformEquicontinuous.equicontinuous
theorem UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
    Equicontinuous F 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} (h : UniformEquicontinuous F)
Goal: Equicontinuous F
-- UniformEquicontinuousOn.equicontinuousOn
theorem UniformEquicontinuousOn.equicontinuousOn {F : ι → β → α} {S : Set β}
    (h : UniformEquicontinuousOn F S) :
    EquicontinuousOn F S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S)
Goal: EquicontinuousOn F S
-- EquicontinuousAt.continuousAt
theorem EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
    ContinuousAt (F i) x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι)
Goal: ContinuousAt (F i) x₀
-- EquicontinuousWithinAt.continuousWithinAt
theorem EquicontinuousWithinAt.continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X}
    (h : EquicontinuousWithinAt F S x₀) (i : ι) :
    ContinuousWithinAt (F i) S x₀ 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} {x₀ : X} (h : EquicontinuousWithinAt F S x₀) (i : ι)
Goal: ContinuousWithinAt (F i) S x₀
-- Set.EquicontinuousAt.continuousAt_of_mem
protected theorem Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
    (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H : Set (X → α)} {x₀ : X} (h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H)
Goal: ContinuousAt f x₀
-- Set.EquicontinuousWithinAt.continuousWithinAt_of_mem
protected theorem Set.EquicontinuousWithinAt.continuousWithinAt_of_mem {H : Set <| X → α}
    {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) {f : X → α} (hf : f ∈ H) :
    ContinuousWithinAt f S x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H : Set (X → α)} {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) {f : X → α} (hf : f ∈ H)
Goal: ContinuousWithinAt f S x₀
-- Equicontinuous.continuous
theorem Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
    Continuous (F i) 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} (h : Equicontinuous F) (i : ι)
Goal: Continuous (F i)
-- EquicontinuousOn.continuousOn
theorem EquicontinuousOn.continuousOn {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S)
    (i : ι) : ContinuousOn (F i) S 
Arguments: {ι : Type u_1} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (i : ι)
Goal: ContinuousOn (F i) S
-- Set.Equicontinuous.continuous_of_mem
protected theorem Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
    {f : X → α} (hf : f ∈ H) : Continuous f 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H : Set (X → α)} (h : H.Equicontinuous) {f : X → α} (hf : f ∈ H)
Goal: Continuous f
-- Set.EquicontinuousOn.continuousOn_of_mem
protected theorem Set.EquicontinuousOn.continuousOn_of_mem {H : Set <| X → α} {S : Set X}
    (h : H.EquicontinuousOn S) {f : X → α} (hf : f ∈ H) : ContinuousOn f S 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H : Set (X → α)} {S : Set X} (h : H.EquicontinuousOn S) {f : X → α} (hf : f ∈ H)
Goal: ContinuousOn f S
-- UniformEquicontinuous.uniformContinuous
theorem UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
    (i : ι) : UniformContinuous (F i) 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} (h : UniformEquicontinuous F) (i : ι)
Goal: UniformContinuous (F i)
-- UniformEquicontinuousOn.uniformContinuousOn
theorem UniformEquicontinuousOn.uniformContinuousOn {F : ι → β → α} {S : Set β}
    (h : UniformEquicontinuousOn F S) (i : ι) :
    UniformContinuousOn (F i) S 
Arguments: {ι : Type u_1} {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S) (i : ι)
Goal: UniformContinuousOn (F i) S
-- Set.UniformEquicontinuous.uniformContinuous_of_mem
protected theorem Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
    (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {H : Set (β → α)} (h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H)
Goal: UniformContinuous f
-- Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem
protected theorem Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem {H : Set <| β → α}
    {S : Set β} (h : H.UniformEquicontinuousOn S) {f : β → α} (hf : f ∈ H) :
    UniformContinuousOn f S 
Arguments: {α : Type u_7} {β : Type u_9} [uα : UniformSpace α] [uβ : UniformSpace β] {H : Set (β → α)} {S : Set β} (h : H.UniformEquicontinuousOn S) {f : β → α} (hf : f ∈ H)
Goal: UniformContinuousOn f S
-- EquicontinuousAt.comp
theorem EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
    EquicontinuousAt (F ∘ u) x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι)
Goal: EquicontinuousAt (F ∘ u) x₀
-- EquicontinuousWithinAt.comp
theorem EquicontinuousWithinAt.comp {F : ι → X → α} {S : Set X} {x₀ : X}
    (h : EquicontinuousWithinAt F S x₀) (u : κ → ι) :
    EquicontinuousWithinAt (F ∘ u) S x₀ 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} {x₀ : X} (h : EquicontinuousWithinAt F S x₀) (u : κ → ι)
Goal: EquicontinuousWithinAt (F ∘ u) S x₀
-- Set.EquicontinuousAt.mono
protected theorem Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
    (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H H' : Set (X → α)} {x₀ : X} (h : H.EquicontinuousAt x₀) (hH : H' ⊆ H)
Goal: H'.EquicontinuousAt x₀
-- Set.EquicontinuousWithinAt.mono
protected theorem Set.EquicontinuousWithinAt.mono {H H' : Set <| X → α} {S : Set X} {x₀ : X}
    (h : H.EquicontinuousWithinAt S x₀) (hH : H' ⊆ H) : H'.EquicontinuousWithinAt S x₀ 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H H' : Set (X → α)} {S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) (hH : H' ⊆ H)
Goal: H'.EquicontinuousWithinAt S x₀
-- Equicontinuous.comp
theorem Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
    Equicontinuous (F ∘ u) 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι)
Goal: Equicontinuous (F ∘ u)
-- EquicontinuousOn.comp
theorem EquicontinuousOn.comp {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (u : κ → ι) :
    EquicontinuousOn (F ∘ u) S 
Arguments: {ι : Type u_1} {κ : Type u_2} {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (u : κ → ι)
Goal: EquicontinuousOn (F ∘ u) S
-- Set.Equicontinuous.mono
protected theorem Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
    (hH : H' ⊆ H) : H'.Equicontinuous 
Arguments: {X : Type u_3} {α : Type u_7} [tX : TopologicalSpace X] [uα : UniformSpace α] {H H' : Set (X → α)} (h : H.Equicontinuous) (hH : H' ⊆ H)
Goal: H'.Equicontinuous