awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
t : Set α
ht_1 : TotallyBounded t
U : SetRel α α
hU : U ∈ 𝓤 α
u : Set α
hu : u.Finite
ht : t ⊆ ⋃ y ∈ u, {x : α | (x, y) ∈ U}
⊢ ∃ (t_1 : Set (Set α)),
    t_1.Finite ∧
      ∀ x ⊆ t,
        ∃ (i : Set α) (_ : i ∈ t_1), Membership.mem (γ := Set (Set α)) {x : Set α | (x, i) ∈ hausdorffEntourage U} x
have refine : ∀ (s : Set α) (hs : s ⊆ t) (x : α) (hx : x ∈ (s, u ∩ U.image s).1), x ∈ U.preimage (s, u ∩ U.image s).2
Tactic: refine ⟨u.powerset, hu.powerset, fun s hs => ⟨u ∩ U.image s, by grind, fun x hx => ?_,
    fun x ⟨_, hx⟩ => hx⟩⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
t : Set α
ht_1 : TotallyBounded t
U : SetRel α α
hU : U ∈ 𝓤 α
u : Set α
hu : u.Finite
ht : t ⊆ ⋃ y ∈ u, {x : α | (x, y) ∈ U}
s : Set α
hs : s ⊆ t
x : α
hx : x ∈ (s, u ∩ U.image s).1
⊢ x ∈ U.preimage (s, u ∩ U.image s).2
have obtain : ∀ (y : α) (hy : y ∈ u) (hxy : Membership.mem (γ := Set α) {x : α | (x, y) ∈ U} x), x ∈ U.preimage (s, u ∩ U.image s).2
Tactic: obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht (hs hx))
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
⊢ 𝓝 s ≤ 𝓝 s
have simp_rw : ∀ (i' : SetRel α α),
  i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a} ∩ {a : Set α | a ⊆ i'.image s} ∈ 𝓝 s
Tactic: simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
⊢ U ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] U → {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
have this : ∀ (hU₁ : U ∈ 𝓤 α) (hU₂ : IsOpen[instTopologicalSpaceProd] U),
  {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
Tactic: ⟨hU₁, hU₂⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (hypothesis)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
⊢ {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
have this : U.IsRefl
Tactic: have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
⊢ {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
have refine : {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
Tactic: refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
⊢ {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
have obtain : ∀ (V : SetRel α α) (hV₁ : V ∈ 𝓤 α) (hV₂ : IsOpen[instTopologicalSpaceProd] V),
  V.IsSymm → ∀ (hVU : V.comp V ⊆ U), {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
Tactic: obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
⊢ {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
have obtain : ∀ (t : Finset α) (ht₁ : ((↑t : Set α), s).1 ⊆ V.preimage ((↑t : Set α), s).2)
  (ht₂ : ((↑t : Set α), s).2 ⊆ V.image ((↑t : Set α), s).1), {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
Tactic: obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
⊢ (∀ i ∈ t, (u ∩ UniformSpace.ball i V).Nonempty) → s ⊆ U.preimage u
have h : ∀ (hu : (↑t : Set α) ⊆ V.preimage u), s ⊆ U.preimage u
Tactic: (hu : ↑t ⊆ V.preimage ↑u)
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
⊢ {a : Set α | s ⊆ U.preimage a} ∈ 𝓝 s
have h : ∀ (u : Set α) (hu : (↑t : Set α) ⊆ V.preimage u), s ⊆ U.preimage u
Tactic: filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
hu : (↑t : Set α) ⊆ V.preimage u
⊢ s ⊆ U.preimage u
have h : V.image (↑t : Set α) ⊆ U.preimage u
Tactic: grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
hu : (↑t : Set α) ⊆ V.preimage u
h : V.image (↑t : Set α) ⊆ U.preimage u
⊢ s ⊆ U.preimage u
have h₁ : V.preimage (↑t : Set α) ⊆ U.preimage u
Tactic: grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
hu : (↑t : Set α) ⊆ V.preimage u
h : V.image (↑t : Set α) ⊆ U.preimage u
h₁ : V.preimage (↑t : Set α) ⊆ U.preimage u
⊢ s ⊆ U.preimage u
have h₂ : V.preimage (V.preimage u) ⊆ U.preimage u
Tactic: grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
hu : (↑t : Set α) ⊆ V.preimage u
h : V.image (↑t : Set α) ⊆ U.preimage u
h₁ : V.preimage (↑t : Set α) ⊆ U.preimage u
h₂ : V.preimage (V.preimage u) ⊆ U.preimage u
⊢ s ⊆ U.preimage u
have h₃ : V.preimage (V.preimage u) ⊆ (V.comp V).preimage u
Tactic: grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
this_1 : U.IsRefl
this : TopologicalSpace (Set α) := TopologicalSpace.vietoris α
V : SetRel α α
hV₁ : V ∈ 𝓤 α
hV₂ : IsOpen[instTopologicalSpaceProd] V
left : V.IsSymm
hVU : V.comp V ⊆ U
t : Finset α
ht₁ : (↑t : Set α) ⊆ V.preimage s
ht₂ : s ⊆ V.image (↑t : Set α)
u : Set α
hu : (↑t : Set α) ⊆ V.preimage u
h : V.image (↑t : Set α) ⊆ U.preimage u
h₁ : V.preimage (↑t : Set α) ⊆ U.preimage u
h₂ : V.preimage (V.preimage u) ⊆ U.preimage u
h₃ : V.preimage (V.preimage u) ⊆ (V.comp V).preimage u
⊢ s ⊆ U.preimage u
have h₄ : V.preimage (V.preimage u) ⊆ V.preimage (V.preimage u)
Tactic: grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
⊢ 𝓝 s ≤ 𝓝 s
have open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
⊢ 𝓝 s ≤ 𝓝 s
have open₁ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₃ : ∀ (i' : SetRel α α),
  i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
    Prod.mk s ⁻¹' {x : Set α × Set α | x.1 ⊆ i'.preimage x.2 ∧ x.2 ⊆ i'.image x.1} ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
open₃ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      Prod.mk s ⁻¹' {x : Set α × Set α | x.1 ⊆ i'.preimage x.2 ∧ x.2 ⊆ i'.image x.1} ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₄ : ∀ (i' : SetRel α α),
  i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a ∧ a ⊆ i'.image s} ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
open₃ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      Prod.mk s ⁻¹' {x : Set α × Set α | x.1 ⊆ i'.preimage x.2 ∧ x.2 ⊆ i'.image x.1} ∈ 𝓝 s
open₄ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a ∧ a ⊆ i'.image s} ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₅ : ∀ (i' : SetRel α α),
  i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a} ∩ {a : Set α | a ⊆ i'.image s} ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
open₃ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      Prod.mk s ⁻¹' {x : Set α × Set α | x.1 ⊆ i'.preimage x.2 ∧ x.2 ⊆ i'.image x.1} ∈ 𝓝 s
open₄ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a ∧ a ⊆ i'.image s} ∈ 𝓝 s
open₅ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      {a : Set α | s ⊆ i'.preimage a} ∩ {a : Set α | a ⊆ i'.image s} ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₆ : ∀ (U : SetRel α α) (hU₁ : U ∈ 𝓤 α) (hU₂ : IsOpen[instTopologicalSpaceProd] U),
  {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]
-- Lean 4 proof step from Mathlib.Topology.UniformSpace.Closeds (goal)
Goal:
α : Type u_1
inst : UniformSpace α
s : Set α
hs : TotallyBounded s
open : 𝓝 s ≤ Filter.comap (Prod.mk s) (𝓤 (Set α))
open₁ :
  ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i' ∈ 𝓝 s
open₂ : ∀ (i' : SetRel α α), i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → Prod.mk s ⁻¹' hausdorffEntourage i' ∈ 𝓝 s
open₃ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      Prod.mk s ⁻¹' {x : Set α × Set α | x.1 ⊆ i'.preimage x.2 ∧ x.2 ⊆ i'.image x.1} ∈ 𝓝 s
open₄ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a : Set α | s ⊆ i'.preimage a ∧ a ⊆ i'.image s} ∈ 𝓝 s
open₅ :
  ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' →
      {a : Set α | s ⊆ i'.preimage a} ∩ {a : Set α | a ⊆ i'.image s} ∈ 𝓝 s
open₆ :
  ∀ (U : SetRel α α) (hU₁ : U ∈ 𝓤 α) (hU₂ : IsOpen[instTopologicalSpaceProd] U),
    {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
⊢ 𝓝 s ≤ 𝓝 s
have open₇ : ∀ (U : SetRel α α) (hU₁ : U ∈ 𝓤 α) (hU₂ : IsOpen[instTopologicalSpaceProd] U) (this : U.IsRefl),
  {a : Set α | s ⊆ U.preimage a} ∩ {a : Set α | a ⊆ U.image s} ∈ 𝓝 s
Tactic: open UniformSpace TopologicalSpace.vietoris in
  simp_rw [nhds_eq_comap_uniformity,
    uniformity_hasBasis_open.uniformity_hausdorff |>.comap _ |>.ge_iff, Function.comp_id,
    hausdorffEntourage, Set.preimage_setOf_eq, Set.setOf_and]
  intro U ⟨hU₁, hU₂⟩
  have : U.IsRefl := ⟨fun _ => refl_mem_uniformity hU₁⟩
  let := TopologicalSpace.vietoris α
  refine Filter.inter_mem ?_ <| hU₂.relImage.powerset_vietoris.mem_nhds <|
    SetRel.self_subset_image _
  obtain ⟨V : SetRel α α, hV₁, hV₂, _, hVU⟩ := comp_open_symm_mem_uniformity_sets hU₁
  obtain ⟨t, ht₁, ht₂⟩ := hs.exists_prodMk_finset_mem_hausdorffEntourage hV₁
  dsimp only at ht₁ ht₂
  filter_upwards [(Filter.eventually_all_finset t).mpr fun x hx =>
    isOpen_inter_nonempty_of_isOpen (isOpen_ball x hV₂) |>.eventually_mem (ht₁ hx)]
    with u (hu : ↑t ⊆ V.preimage ↑u)
  grw [ht₂, ← SetRel.preimage_eq_image, hu, ← hVU, SetRel.preimage_comp]