-- 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
g : F.EtaleSpace (CC := CC) (FC := FC)
hg :
  Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
    (base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) g
rcases :
  ∀ (f : CC (F.obj (op U)))
    (hf :
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g.base (CC := CC) (FC := FC)) hg) :
            (a : CC (F.obj (op U))) → CC (F.stalk g.base))
          f =
        g.germ (CC := CC) (FC := FC)),
    ∀ᶠ (b : F.EtaleSpace (CC := CC) (FC := FC)) in
      𝓝
        (↑(Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
              Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
                (base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
              g hg) :
          F.EtaleSpace),
      ∀ (a : (↑(base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) : Type v)),
        (↑a : F.EtaleSpace) = b →
          Eq (α := WithDiscreteTopology (ToType (FC := FC) (CC := CC) (F.stalk x)))
            (WithTopology.toTopology ⊥
                ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
                  (surjInv ⋯ ((↑a : F.EtaleSpace).germ (CC := CC) (FC := FC)))) :
              WithTopology (ToType (FC := FC) (CC := CC) (F.stalk x)) ⊥)
            (WithTopology.toTopology ⊥
                ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
                  (surjInv ⋯
                    ((↑(Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
                              Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
                                (base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
                              g hg) :
                          F.EtaleSpace).germ
                      (CC := CC) (FC := FC)))) :
              WithTopology (ToType (FC := FC) (CC := CC) (F.stalk x)) ⊥)
⊢ ∀ᶠ (b : F.EtaleSpace (CC := CC) (FC := FC)) in
    𝓝
      (↑(Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
            Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
              (base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
            g hg) :
        F.EtaleSpace),
    ∀ (a : (↑(base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) : Type v)),
      (↑a : F.EtaleSpace) = b →
        Eq (α := WithDiscreteTopology (ToType (FC := FC) (CC := CC) (F.stalk x)))
          (WithTopology.toTopology ⊥
              ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
                (surjInv ⋯ ((↑a : F.EtaleSpace).germ (CC := CC) (FC := FC)))) :
            WithTopology (ToType (FC := FC) (CC := CC) (F.stalk x)) ⊥)
          (WithTopology.toTopology ⊥
              ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
                (surjInv ⋯
                  ((↑(Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
                            Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
                              (base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
                            g hg) :
                        F.EtaleSpace).germ
                    (CC := CC) (FC := FC)))) :
            WithTopology (ToType (FC := FC) (CC := CC) (F.stalk x)) ⊥)

Tactic:
rcases hF_bij _ hg |>.surjective g.germ with ⟨f, hf⟩

Premises used: Function.Bijective.surjective, TopCat.Presheaf.EtaleSpace.germ