-- Lean 4 proof step from Mathlib.Topology.Sheaves.LocalPredicate (goal)
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
x : (↑X : Type u_1)
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))
hU : P.toPrelocalPredicate (T := T).pred (T := T) fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))
hV : P.toPrelocalPredicate (T := T).pred (T := T) fV
h :
  Eq (α := T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app U) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
      (P.cocone (T := T) x).pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app V) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
      (P.cocone (T := T) x).pt)
w :
  ∃ (W : OpenNhds x) (iU : W ⟶ unop U) (iV : W ⟶ unop V),
    ∀ (w : ↥(↑W : Opens (↑X : Type u_1))),
      Eq (α :=
        T (↑((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w) : (↑X : Type u_1)))
        (fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w))
        (fV ((iV : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop V) : Opens (↑X : Type u_1))) w))
⊢ ∃ (W : (OpenNhds x)ᵒᵖ) (s : (w : ↥(↑(unop W) : Opens (↑X : Type u_1))) → T (↑w : (↑X : Type u_1))) (hW :
    P.toPrelocalPredicate (T := T).pred (T := T) s),
    Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (colimit.ι
                  (Functor.obj
                    (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                      (OpenNhds.inclusion x).op)
                    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                  U) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                              (OpenNhds.inclusion x).op).obj
                          (subpresheafToTypes P.toPrelocalPredicate)).obj
                      U)) →
                (fun (X : Type u_1) ↦ X)
                  (colimit
                    (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                          (OpenNhds.inclusion x).op).obj
                      (subpresheafToTypes P.toPrelocalPredicate))))
            (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
          (fun (X : Type u_1) ↦ X)
            (colimit
              (Functor.obj
                (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                  (OpenNhds.inclusion x).op)
                (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x ⋯) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((subsheafToTypes P).presheaf.obj (op (↑(unop W) : Opens (↑X : Type u_1))))) →
                (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
            (Subtype.mk (p :=
              fun (f : (x_1 : ↥(unop (op (↑(unop W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) :
          (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)) ∧
      Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (colimit.ι
                  (Functor.obj
                    (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                      (OpenNhds.inclusion x).op)
                    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                  V) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                              (OpenNhds.inclusion x).op).obj
                          (subpresheafToTypes P.toPrelocalPredicate)).obj
                      V)) →
                (fun (X : Type u_1) ↦ X)
                  (colimit
                    (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                          (OpenNhds.inclusion x).op).obj
                      (subpresheafToTypes P.toPrelocalPredicate))))
            (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
          (fun (X : Type u_1) ↦ X)
            (colimit
              (Functor.obj
                (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                  (OpenNhds.inclusion x).op)
                (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x ⋯) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((subsheafToTypes P).presheaf.obj (op (↑(unop W) : Opens (↑X : Type u_1))))) →
                (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
            (Subtype.mk (p :=
              fun (f : (x_1 : ↥(unop (op (↑(unop W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) :
          (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
have refine_1 : ∀ (W : OpenNhds x) (iU : W ⟶ unop U) (iV : W ⟶ unop V)
  (w :
    ∀ (w : ↥(↑W : Opens (↑X : Type u_1))),
      Eq (α :=
        T (↑((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w) : (↑X : Type u_1)))
        (fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop U) : Opens (↑X : Type u_1))) w))
        (fV ((iV : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑(unop V) : Opens (↑X : Type u_1))) w))),
  ∃ (W : (OpenNhds x)ᵒᵖ) (s : (w : ↥(↑(unop W) : Opens (↑X : Type u_1))) → T (↑w : (↑X : Type u_1))) (hW :
    P.toPrelocalPredicate (T := T).pred (T := T) s),
    Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (colimit.ι
                  (Functor.obj
                    (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                      (OpenNhds.inclusion x).op)
                    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                  U) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                              (OpenNhds.inclusion x).op).obj
                          (subpresheafToTypes P.toPrelocalPredicate)).obj
                      U)) →
                (fun (X : Type u_1) ↦ X)
                  (colimit
                    (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                          (OpenNhds.inclusion x).op).obj
                      (subpresheafToTypes P.toPrelocalPredicate))))
            (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
          (fun (X : Type u_1) ↦ X)
            (colimit
              (Functor.obj
                (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                  (OpenNhds.inclusion x).op)
                (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x sorry) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((subsheafToTypes P).presheaf.obj (op (↑(unop W) : Opens (↑X : Type u_1))))) →
                (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
            (Subtype.mk (p :=
              fun (f : (x_1 : ↥(unop (op (↑(unop W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) :
          (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x)) ∧
      Eq (α := (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (colimit.ι
                  (Functor.obj
                    (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                      (OpenNhds.inclusion x).op)
                    (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                  V) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                              (OpenNhds.inclusion x).op).obj
                          (subpresheafToTypes P.toPrelocalPredicate)).obj
                      V)) →
                (fun (X : Type u_1) ↦ X)
                  (colimit
                    (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                          (OpenNhds.inclusion x).op).obj
                      (subpresheafToTypes P.toPrelocalPredicate))))
            (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
          (fun (X : Type u_1) ↦ X)
            (colimit
              (Functor.obj
                (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                  (OpenNhds.inclusion x).op)
                (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                ((subsheafToTypes (T := T) P).presheaf.germ (↑(unop W) : Opens (↑X : Type u_1)) x sorry) :
              (a :
                  (fun (X : Type u_1) ↦ X)
                    ((subsheafToTypes P).presheaf.obj (op (↑(unop W) : Opens (↑X : Type u_1))))) →
                (fun (X : Type u_1) ↦ X) ((subsheafToTypes P).presheaf.stalk x))
            (Subtype.mk (p :=
              fun (f : (x_1 : ↥(unop (op (↑(unop W) : Opens (↑X : Type u_1))))) → T (↑x_1 : (↑X : Type u_1))) ↦
              P.toPrelocalPredicate (T := T).pred (T := T) f) s hW) :
          (fun (X : Type u_1) ↦ X) ((subsheafToTypes (T := T) P).presheaf.stalk x))
Tactic: rcases w with ⟨W, iU, iV, w⟩
Cleared hypotheses: w