l3lab/ntp-mathlib huggingface dataset:
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by 

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
⊢ ⋃ a, (fun r => ↑r) a = univ

Next tactic:
rwa [← sUnion_range, Subtype.range_coe]
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
this :
  ∃ β t, ∃ (_ : ∀ (b : β), IsOpen (t b)) (_ : ⋃ b, t b = univ), LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ (fun r => ↑r) a
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
this : ∃ β t, (∀ (b : β), IsOpen (t b)) ∧ (∀ (x : X), ∃ i, x ∈ t i) ∧ LocallyFinite t ∧ ∀ (b : β), ∃ i, t b ⊆ u i
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
choose α t hto hXt htf ind hind using this
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
hXt : ∀ (x : X), ∃ i, x ∈ t i
htf : LocallyFinite t
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
choose t_inv ht_inv using hXt
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
htf : LocallyFinite t
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
choose U hxU hU using htf
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  

Proof state:
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a

Next tactic:
refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · 

Proof state:
case refine'_1
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ ∀ (a : ι), IsOpen ((fun i => ⋃ a, ⋃ (_ : ind a = i), t a) a)

Next tactic:
exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · 

Proof state:
case refine'_2
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ ⋃ i, (fun i => ⋃ a, ⋃ (_ : ind a = i), t a) i = univ

Next tactic:
simp only [eq_univ_iff_forall, mem_iUnion]
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    

Proof state:
case refine'_2
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ ∀ (x : X), ∃ i i_1, ∃ (_ : ind i_1 = i), x ∈ t i_1

Next tactic:
exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
  · 

Proof state:
case refine'_3
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ LocallyFinite fun i => ⋃ a, ⋃ (_ : ind a = i), t a

Next tactic:
refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
  · refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
    

Proof state:
case refine'_3
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
x : X
⊢ {i | Set.Nonempty ((fun i => ⋃ a, ⋃ (_ : ind a = i), t a) i ∩ U x)} ⊆ ind '' {i | Set.Nonempty (t i ∩ U x)}

Next tactic:
simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
  · refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
    

Proof state:
case refine'_3
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
x : X
⊢ ∀ (x_1 : ι), (∃ x_2, (∃ i, ∃ (_ : ind i = x_1), x_2 ∈ t i) ∧ x_2 ∈ U x) → x_1 ∈ ind '' {i | ∃ x_2 ∈ t i, x_2 ∈ U x}

Next tactic:
rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
  · refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
    rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩
    

Proof state:
case refine'_3.intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
x y : X
hyU : y ∈ U x
a : α
hya : y ∈ t a
⊢ ind a ∈ ind '' {i | ∃ x_1 ∈ t i, x_1 ∈ U x}

Next tactic:
exact mem_image_of_mem _ ⟨y, hya, hyU⟩
/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
one indexed on the same type with each open set contained in the corresponding original one. -/
theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
  choose α t hto hXt htf ind hind using this
  choose t_inv ht_inv using hXt
  choose U hxU hU using htf
  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
  · simp only [eq_univ_iff_forall, mem_iUnion]
    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
  · refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
    rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩
    exact mem_image_of_mem _ ⟨y, hya, hyU⟩
  · 

Proof state:
case refine'_4
ι : Type u
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace X
u : ι → Set X
uo : ∀ (a : ι), IsOpen (u a)
uc : ⋃ i, u i = univ
α : Type v
t : α → Set X
hto : ∀ (b : α), IsOpen (t b)
ind : α → ι
hind : ∀ (b : α), t b ⊆ u (ind b)
t_inv : X → α
ht_inv : ∀ (x : X), x ∈ t (t_inv x)
U : X → Set X
hxU : ∀ (x : X), U x ∈ 𝓝 x
hU : ∀ (x : X), Set.Finite {i | Set.Nonempty (t i ∩ U x)}
⊢ ∀ (a : ι), (fun i => ⋃ a, ⋃ (_ : ind a = i), t a) a ⊆ u a

Next tactic:
simp only [subset_def, mem_iUnion]