l3lab/ntp-mathlib huggingface dataset:
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
⊢ ∃ ι x C,
    ∃ (_ : ∀ (i : ι), IsCompact (C i)),
      ∃ U,
        ∃ (_ : ∀ (i : ι), IsOpen (U i)),
          f ∈ ⋂ i, CompactOpen.gen (C i) (U i) ∧ ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f

Next tactic:
have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
⊢ K ⊆ ⋃ i, C i

Next tactic:
rw [← K.inter_iUnion, subset_inter_iff]
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
⊢ K ⊆ K ∧ K ⊆ ⋃ i, closure (U ↑↑i)

Next tactic:
refine' ⟨Subset.rfl, ht.trans _⟩
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
⊢ ⋃ i ∈ t, U ↑i ⊆ ⋃ i, closure (U ↑↑i)

Next tactic:
simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
⊢ ∀ (x : α) (h : x ∈ K), { val := x, property := h } ∈ t → ⇑f ⁻¹' ball (f x) Z ⊆ ⋃ i, closure (⇑f ⁻¹' ball (f ↑↑i) Z)

Next tactic:
exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by 

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
x : α
hx₁ : x ∈ K
hx₂ : { val := x, property := hx₁ } ∈ t
⊢ ⇑f ⁻¹' ball (f x) Z ⊆ closure (⇑f ⁻¹' ball (f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)

Next tactic:
simp [subset_closure]
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
hC : K ⊆ ⋃ i, C i
⊢ ∃ ι x C,
    ∃ (_ : ∀ (i : ι), IsCompact (C i)),
      ∃ U,
        ∃ (_ : ∀ (i : ι), IsOpen (U i)),
          f ∈ ⋂ i, CompactOpen.gen (C i) (U i) ∧ ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f

Next tactic:
have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
    simp only [← image_subset_iff, ← mem_preimage]
    rintro ⟨⟨x, hx₁⟩, hx₂⟩
    have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by
      intro y hy
      obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁
      exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))
    calc
      f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)
      _ ⊆ closure (f '' U x) := f.continuous.continuousOn.image_closure
      _ ⊆ closure (ball (f x) Z) := by
        apply closure_mono
        simp only [image_subset_iff]
        rfl
      _ ⊆ ball (f x) W := hZW
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
hC : K ⊆ ⋃ i, C i
⊢ ∀ (i : { x // x ∈ t }), C i ⊆ ⇑f ⁻¹' ball (f ↑↑i) W

Next tactic:
simp only [← image_subset_iff, ← mem_preimage]
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
    simp only [← image_subset_iff, ← mem_preimage]
    

Proof state:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
hC : K ⊆ ⋃ i, C i
⊢ ∀ (i : { x // x ∈ t }), ⇑f '' (K ∩ closure (⇑f ⁻¹' ball (f ↑↑i) Z)) ⊆ ball (f ↑↑i) W

Next tactic:
rintro ⟨⟨x, hx₁⟩, hx₂⟩
/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
compact-open topology.

Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
that the compact-open topology is at least as fine as the topology of compact convergence. -/
theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
    intro x hx
    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
  have hC : K ⊆ ⋃ i, C i := by
    rw [← K.inter_iUnion, subset_inter_iff]
    refine' ⟨Subset.rfl, ht.trans _⟩
    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
    simp only [← image_subset_iff, ← mem_preimage]
    rintro ⟨⟨x, hx₁⟩, hx₂⟩
    

Proof state:
case mk.mk
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
hK : IsCompact K
hV : V ∈ 𝓤 β
W : Set (β × β)
hW₁ : W ∈ 𝓤 β
hW₄ : IsOpen W
hW₂ : SymmetricRel W
hW₃ : W ○ W ⊆ V
Z : Set (β × β)
hZ₁ : Z ∈ 𝓤 β
hZ₄ : IsOpen Z
hZ₂ : SymmetricRel Z
hZ₃ : Z ○ Z ⊆ W
U : α → Set α := fun x => ⇑f ⁻¹' ball (f x) Z
hU : ∀ (x : α), IsOpen (U x)
hUK : K ⊆ ⋃ x, U ↑x
t : Finset ↑K
ht : K ⊆ ⋃ i ∈ t, U ↑i
C : { x // x ∈ t } → Set α := fun i => K ∩ closure (U ↑↑i)
hC : K ⊆ ⋃ i, C i
x : α
hx₁ : x ∈ K
hx₂ : { val := x, property := hx₁ } ∈ t
⊢ ⇑f '' (K ∩ closure (⇑f ⁻¹' ball (f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) Z)) ⊆
    ball (f ↑↑{ val := { val := x, property := hx₁ }, property := hx₂ }) W

Next tactic:
have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by
      intro y hy
      obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁
      exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))