-- Module: Mathlib.Topology.Sheaves.EtaleSpace
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
U : Opens (↑X : Type v)
hF_bij :
  ∀ (x : (↑X : Type v)) (hx : x ∈ U),
    Bijective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (F.stalk x))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx)) : (a : CC (F.obj (op U))) → CC (F.stalk x))
x : (↑X : Type v)
hx : x ∈ U
base : (↑X : Type v)
s : ToType (FC := FC) (CC := CC) (F.stalk base)
hs :
  Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
    (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) (mk (CC := CC) (FC := FC) base s)
simp :
  Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
      (mk (CC := CC) (FC := FC) base
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base ⋯) : (a : CC (F.obj (op U))) → CC (F.stalk base))
          (surjInv ⋯
            ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
              (surjInv ⋯ s)))))
      ⋯ =
    Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
      (mk (CC := CC) (FC := FC) base s) hs
⊢ (fun (x_1 : ↥U × WithDiscreteTopology (ToType (FC := FC) (CC := CC) (F.stalk x))) ↦
        (match (motive :=
            ↥U × WithDiscreteTopology (ToType (FC := FC) (CC := CC) (F.stalk x)) →
              (↑(EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) : Type v))
            x_1 with
          | (⟨y, hy⟩, WithTopology.toTopology .(⊥) g) =>
            Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
              Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
                (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
              (mk (CC := CC) (FC := FC) y
                ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U y hy) : (a : CC (F.obj (op U))) → CC (F.stalk y))
                  (surjInv ⋯ g)))
              hy :
          (↑(EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) : Type v)))
      ((fun (s : (↑(EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) : Type v)) ↦
          ((Subtype.mk (p := fun (x : (↑X : Type v)) ↦ x ∈ U) ((↑s : F.EtaleSpace).base (CC := CC) (FC := FC)) ⋯,
              WithTopology.toTopology ⊥
                ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
                  (surjInv ⋯ ((↑s : F.EtaleSpace).germ (CC := CC) (FC := FC))))) :
            ↥U × WithDiscreteTopology (ToType (FC := FC) (CC := CC) (F.stalk x))))
        (Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
          Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
            (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
          (mk (CC := CC) (FC := FC) base s) hs)) =
    Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
      (mk (CC := CC) (FC := FC) base s) hs

Tactic:
simp only