-- 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
as : ι
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))
as_1 : ι
x : (↑(↑((F ⋙ forgetToPresheafedSpace).obj { as := as_1 : Discrete ι }) : TopCat) : Type v)
h : { as := as : Discrete ι } ≠ { as := as_1 }
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) (X :=
            Functor.obj (forget C) (F.obj { as := as_1 : Discrete ι }))
            (CategoryStruct.comp (Y := Functor.obj (forget C) (colimit F))
              (CategoryTheory.Functor.map (forget C) (colimit.ι F { as := as_1 }))
              (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_1 })) : 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))
⊢ Eq (α := (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))
    (((@ConcreteCategory.hom _ TopCat.instCategory (fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v)))
              TopCat.carrier _ _ (Functor.obj (forget C) (F.obj { as := as : Discrete ι }))
              (TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v)))
              (NatTrans.app (Iso.hom Discrete.natIsoFunctor) { as := as } ≫
                  TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) as :
                (F ⋙ forget C).obj { as := as : Discrete ι } ⟶
                  TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) :
            C((↑(Functor.obj (forget C) (F.obj { as := as : Discrete ι })) : Type v),
              (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : 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 _ TopCat.instCategory (fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v)))
              TopCat.carrier _ _ (Functor.obj (forget C) (F.obj { as := as_1 : Discrete ι }))
              (TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v)))
              (NatTrans.app (Iso.hom Discrete.natIsoFunctor) { as := as_1 } ≫
                  TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) as_1 :
                (F ⋙ forget C).obj { as := as_1 : Discrete ι } ⟶
                  TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) :
            C((↑(Functor.obj (forget C) (F.obj { as := as_1 : Discrete ι })) : Type v),
              (↑(TopCat.of ((i : ι) × (↑(((F ⋙ forget C).obj ∘ Discrete.mk) i) : Type v))) : Type v))) :
          (a : (↑((forget C).obj (F.obj { as := as_1 })) : 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))

Tactic:
rw [ι_preservesColimitIso_hom_assoc, ι_preservesColimitIso_hom_assoc,
    HasColimit.isoOfNatIso_ι_hom_assoc, HasColimit.isoOfNatIso_ι_hom_assoc,
    TopCat.sigmaIsoSigma_hom_ι, TopCat.sigmaIsoSigma_hom_ι] at eq

Premises used: TopCat.sigmaIsoSigma_hom_ι, CategoryTheory.ι_preservesColimitIso_hom_assoc, CategoryTheory.Limits.HasColimit.isoOfNatIso_ι_hom_assoc