-- Module: Mathlib.Geometry.RingedSpace.OpenImmersion
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasLimits.{v, u} C
ι : Type v
F : Discrete ι ⥤ SheafedSpace.{u, v, v} C
inst_2 : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens (↑(↑(F.obj i).toPresheafedSpace : TopCat) : Type v)
x : (↑(↑((F ⋙ forgetToPresheafedSpace).obj j) : TopCat) : Type v)
y : (↑(↑(F.obj i).toPresheafedSpace : TopCat) : Type v)
hy : y ∈ (↑U : Set (↑(↑(F.obj i).toPresheafedSpace : TopCat) : Type v))
eq :
  Eq (α := (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (CategoryStruct.comp (Y := Functor.obj (forget C) (colimit F))
              (CategoryTheory.Functor.map (forget C) (colimit.ι F i))
              (CategoryStruct.comp (Y := colimit (F ⋙ forget C))
                (Iso.hom (preservesColimitIso (forget C) F) :
                  Functor.obj (forget C) (colimit F) ⟶ colimit (F ⋙ forget C))
                (Iso.hom (HasColimit.isoOfNatIso Discrete.natIsoFunctor) ≫
                    (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).hom :
                  colimit (F ⋙ forget C) ⟶ TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))))) :
          (a : (↑((forget C).obj (F.obj i)) : Type v)) →
            (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
        y :
      (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (CategoryStruct.comp (Y := Functor.obj (forget C) (colimit F))
              (CategoryTheory.Functor.map (forget C) (colimit.ι F j))
              (CategoryStruct.comp (Y := colimit (F ⋙ forget C))
                (Iso.hom (preservesColimitIso (forget C) F) :
                  Functor.obj (forget C) (colimit F) ⟶ colimit (F ⋙ forget C))
                (Iso.hom (HasColimit.isoOfNatIso Discrete.natIsoFunctor) ≫
                    (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).hom :
                  colimit (F ⋙ forget C) ⟶ TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))))) :
          (a : (↑((forget C).obj (F.obj j)) : Type v)) →
            (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
        x :
      (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
h₁ :
  ∀ (as : ι) (h : { as := as : Discrete ι } ≠ j)
    (U : Opens (↑(↑(F.obj { as := as : Discrete ι }).toPresheafedSpace : TopCat) : Type v))
    (y : (↑(↑(F.obj { as := as : Discrete ι }).toPresheafedSpace : TopCat) : Type v))
    (hy : y ∈ (↑U : Set (↑(↑(F.obj { as := as }).toPresheafedSpace : TopCat) : Type v)))
    (eq :
      Eq (α := (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := Functor.obj (forget C) (F.obj { as := as : Discrete ι }))
                (CategoryStruct.comp (Y := Functor.obj (forget C) (colimit F))
                  (CategoryTheory.Functor.map (forget C) (colimit.ι F { as := as }))
                  (CategoryStruct.comp (Y := colimit (F ⋙ forget C))
                    (Iso.hom (preservesColimitIso (forget C) F) :
                      Functor.obj (forget C) (colimit F) ⟶ colimit (F ⋙ forget C))
                    (Iso.hom (HasColimit.isoOfNatIso Discrete.natIsoFunctor) ≫
                        (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).hom :
                      colimit (F ⋙ forget C) ⟶
                        TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))))) :
              (a : (↑((forget C).obj (F.obj { as := as })) : Type v)) →
                (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
            y :
          (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (CategoryStruct.comp (Y := Functor.obj (forget C) (colimit F))
                  (CategoryTheory.Functor.map (forget C) (colimit.ι F j))
                  (CategoryStruct.comp (Y := colimit (F ⋙ forget C))
                    (Iso.hom (preservesColimitIso (forget C) F) :
                      Functor.obj (forget C) (colimit F) ⟶ colimit (F ⋙ forget C))
                    (Iso.hom (HasColimit.isoOfNatIso Discrete.natIsoFunctor) ≫
                        (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).hom :
                      colimit (F ⋙ forget C) ⟶
                        TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))))) :
              (a : (↑((forget C).obj (F.obj j)) : Type v)) →
                (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
            x :
          (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))),
    False
⊢ False

Tactic:
cases i