-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
U : (OpenNhds x)ᵒᵖ
op :
  ∀ (U : OpenNhds x)
    (s :
      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
            forget C (FC := FC) (CC := CC)).obj
        (Opposite.op U))
    (e :
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (mapCocone (forget C (FC := FC) (CC := CC))
                    (colimit.cocone
                      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op)
                        F))))
                (Opposite.op U)) :
            (a :
                (fun (X : Type v) ↦ X)
                  ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                        forget C).obj
                    (Opposite.op U))) →
              (fun (X : Type v) ↦ X)
                (((Functor.const (OpenNhds x)ᵒᵖ).obj
                      ((forget C).mapCocone
                          (colimit.cocone
                            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj
                                  (OpenNhds.inclusion x).op).obj
                              F))).pt).obj
                  (Opposite.op U)))
          s =
        t),
    ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (Opposite.op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (Opposite.op U))) → CC (F.stalk x))
          s =
        t
⊢ ∀
    (s :
      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
            forget C (FC := FC) (CC := CC)).obj
        U)
    (e :
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (mapCocone (forget C (FC := FC) (CC := CC))
                    (colimit.cocone
                      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op)
                        F))))
                U) :
            (a :
                (fun (X : Type v) ↦ X)
                  ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                        forget C).obj
                    U)) →
              (fun (X : Type v) ↦ X)
                (((Functor.const (OpenNhds x)ᵒᵖ).obj
                      ((forget C).mapCocone
                          (colimit.cocone
                            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj
                                  (OpenNhds.inclusion x).op).obj
                              F))).pt).obj
                  U))
          s =
        t),
    ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (Opposite.op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (Opposite.op U))) → CC (F.stalk x))
          s =
        t

Tactic:
induction U with | op U => ?_

Premises used: Opposite.op