Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
⊢ ∃! F, F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f

Tactic:
refine ⟨⟨F, continuous_iff_continuousAt.mpr fun a ↦ ?_⟩, ⟨?_, funext this⟩, fun F' ⟨F'_0, hpF'⟩ ↦
    DFunLike.ext _ _ fun a ↦ ?_⟩

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
a : A
⊢ ContinuousAt F a

case refine_2
E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
⊢ { toFun := F, continuous_toFun := ⋯ } a₀ = e₀

case refine_3
E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
F' : C(A, E)
x✝ : (fun F ↦ F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f) F'
F'_0 : F' a₀ = e₀
hpF' : p ∘ ⇑F' = ⇑f
a : A
⊢ F' a = { toFun := F, continuous_toFun := ⋯ } a