l3lab/ntp-mathlib huggingface dataset:
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  

Proof state:
case h.h
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
⊢ x ∈
      ↑((Opens.map (colimit.ι (F ⋙ forgetToPresheafedSpace) j).base).obj
          ((Opens.map (preservesColimitIso forgetToPresheafedSpace F).inv.base).obj
            ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U))) ↔
    x ∈ ↑⊥

Next tactic:
apply iff_false_intro
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  

Proof state:
case h.h.h
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
⊢ x ∉
    ↑((Opens.map (colimit.ι (F ⋙ forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso forgetToPresheafedSpace F).inv.base).obj
          ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U)))

Next tactic:
rintro ⟨y, hy, eq⟩
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  (colimit.ι F i).base y =
    (preservesColimitIso forgetToPresheafedSpace F).inv.base ((colimit.ι (F ⋙ forgetToPresheafedSpace) j).base x)
⊢ False

Next tactic:
replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  (preservesColimitIso (forget C) F ≪≫
          HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫
            TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom
      ((colimit.ι F i).base y) =
    (preservesColimitIso (forget C) F ≪≫
          HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫
            TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom
      ((preservesColimitIso forgetToPresheafedSpace F).inv.base ((colimit.ι (F ⋙ forgetToPresheafedSpace) j).base x))
⊢ False

Next tactic:
simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  ((preservesColimitIso (forget C) F).hom ≫
        (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
          (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      ((colimit.ι F i).base y) =
    ((preservesColimitIso (forget C) F).hom ≫
        (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
          (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      ((colimit.ι (F ⋙ forgetToPresheafedSpace) j ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).base x)
⊢ False

Next tactic:
rw [ι_preservesColimitsIso_inv] at eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  ((preservesColimitIso (forget C) F).hom ≫
        (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
          (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      ((colimit.ι F i).base y) =
    ((preservesColimitIso (forget C) F).hom ≫
        (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
          (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      ((forgetToPresheafedSpace.map (colimit.ι F j)).base x)
⊢ False

Next tactic:
erw [← comp_apply, ← comp_apply] at eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  erw [← comp_apply, ← comp_apply] at eq
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  ((colimit.ι F i).base ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      y =
    ((forgetToPresheafedSpace.map (colimit.ι F j)).base ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      x
⊢ False

Next tactic:
change
    ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =
      ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  erw [← comp_apply, ← comp_apply] at eq
  change
    ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =
      ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq
  

Proof state:
case h.h.h.intro.intro
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i j : Discrete ι
h : i ≠ j
U : Opens ↑↑(F.obj i).toPresheafedSpace
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
y : (CategoryTheory.forget TopCat).obj ↑(F.obj i).toPresheafedSpace
hy : y ∈ ↑U
eq :
  ((forget C).map (colimit.ι F i) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      y =
    ((forget C).map (colimit.ι F j) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      x
⊢ False

Next tactic:
cases i
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  erw [← comp_apply, ← comp_apply] at eq
  change
    ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =
      ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq
  cases i; 

Proof state:
case h.h.h.intro.intro.mk
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
j : Discrete ι
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj j)
as✝ : ι
h : { as := as✝ } ≠ j
U : Opens ↑↑(F.obj { as := as✝ }).toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(F.obj { as := as✝ }).toPresheafedSpace
hy : y ∈ ↑U
eq :
  ((forget C).map (colimit.ι F { as := as✝ }) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      y =
    ((forget C).map (colimit.ι F j) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      x
⊢ False

Next tactic:
cases j
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  erw [← comp_apply, ← comp_apply] at eq
  change
    ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =
      ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq
  cases i; cases j
  

Proof state:
case h.h.h.intro.intro.mk.mk
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
as✝¹ : ι
U : Opens ↑↑(F.obj { as := as✝¹ }).toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(F.obj { as := as✝¹ }).toPresheafedSpace
hy : y ∈ ↑U
as✝ : ι
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj { as := as✝ })
h : { as := as✝¹ } ≠ { as := as✝ }
eq :
  ((forget C).map (colimit.ι F { as := as✝¹ }) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      y =
    ((forget C).map (colimit.ι F { as := as✝ }) ≫
        (preservesColimitIso (forget C) F).hom ≫
          (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom ≫
            (TopCat.sigmaIsoSigma ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk)).hom)
      x
⊢ False

Next tactic:
rw [ι_preservesColimitsIso_hom_assoc, ι_preservesColimitsIso_hom_assoc,
    HasColimit.isoOfNatIso_ι_hom_assoc, HasColimit.isoOfNatIso_ι_hom_assoc,
    TopCat.sigmaIsoSigma_hom_ι, TopCat.sigmaIsoSigma_hom_ι] at eq
theorem image_preimage_is_empty (j : Discrete ι) (h : i ≠ j) (U : Opens (F.obj i)) :
    (Opens.map (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) j).base).obj
        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj
          ((sigma_ι_openEmbedding F i).isOpenMap.functor.obj U)) =
      ⊥ := by
  ext x
  apply iff_false_intro
  rintro ⟨y, hy, eq⟩
  replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
    HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
  simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.comp_base] at eq
  rw [ι_preservesColimitsIso_inv] at eq
  -- Porting note : without this `erw`, change does not work
  erw [← comp_apply, ← comp_apply] at eq
  change
    ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =
      ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq
  cases i; cases j
  rw [ι_preservesColimitsIso_hom_assoc, ι_preservesColimitsIso_hom_assoc,
    HasColimit.isoOfNatIso_ι_hom_assoc, HasColimit.isoOfNatIso_ι_hom_assoc,
    TopCat.sigmaIsoSigma_hom_ι, TopCat.sigmaIsoSigma_hom_ι] at eq
  

Proof state:
case h.h.h.intro.intro.mk.mk
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
as✝¹ : ι
U : Opens ↑↑(F.obj { as := as✝¹ }).toPresheafedSpace
y : (CategoryTheory.forget TopCat).obj ↑(F.obj { as := as✝¹ }).toPresheafedSpace
hy : y ∈ ↑U
as✝ : ι
x : ↑↑((F ⋙ forgetToPresheafedSpace).obj { as := as✝ })
h : { as := as✝¹ } ≠ { as := as✝ }
eq :
  (Discrete.natIsoFunctor.hom.app { as := as✝¹ } ≫ TopCat.sigmaι ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk) as✝¹)
      y =
    (Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ TopCat.sigmaι ((F ⋙ forget C).toPrefunctor.obj ∘ Discrete.mk) as✝) x
⊢ False

Next tactic:
exact h (congr_arg Discrete.mk (congr_arg Sigma.fst eq))
instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    

Proof state:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
⊢ IsIso ((colimit.ι F i).c.app (op ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U)))

Next tactic:
have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
    

Proof state:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
e : colimit.ι F i = colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv
⊢ IsIso ((colimit.ι F i).c.app (op ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U)))

Next tactic:
have H :
      OpenEmbedding
        (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
            (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).base :=
      e ▸ sigma_ι_openEmbedding F i
instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
    have H :
      OpenEmbedding
        (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
            (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).base :=
      e ▸ sigma_ι_openEmbedding F i
    

Proof state:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
e : colimit.ι F i = colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv
H :
  OpenEmbedding ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).base
⊢ IsIso ((colimit.ι F i).c.app (op ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U)))

Next tactic:
suffices IsIso <| (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
        (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).c.app <|
      op (H.isOpenMap.functor.obj U) by
      -- Porting note : just `convert` is very slow, so helps it a bit
      convert this using 2 <;> congr
instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
    have H :
      OpenEmbedding
        (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
            (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).base :=
      e ▸ sigma_ι_openEmbedding F i
    suffices IsIso <| (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
        (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).c.app <|
      op (H.isOpenMap.functor.obj U) by
      -- Porting note : just `convert` is very slow, so helps it a bit
      

Proof state:
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
e : colimit.ι F i = colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv
H :
  OpenEmbedding ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).base
this :
  IsIso
    ((colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).c.app
      (op
        ((IsOpenMap.functor
              (_ :
                IsOpenMap
                  ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                        (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
          U)))
⊢ IsIso ((colimit.ι F i).c.app (op ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U)))

Next tactic:
convert this using 2
instance sigma_ι_isOpenImmersion [HasStrictTerminalObjects C] :
    SheafedSpace.IsOpenImmersion (colimit.ι F i) where
  base_open := sigma_ι_openEmbedding F i
  c_iso U := by
    have e : colimit.ι F i = _ :=
      (ι_preservesColimitsIso_inv SheafedSpace.forgetToPresheafedSpace F i).symm
    have H :
      OpenEmbedding
        (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
            (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).base :=
      e ▸ sigma_ι_openEmbedding F i
    suffices IsIso <| (colimit.ι (F ⋙ SheafedSpace.forgetToPresheafedSpace) i ≫
        (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv).c.app <|
      op (H.isOpenMap.functor.obj U) by
      -- Porting note : just `convert` is very slow, so helps it a bit
      convert this using 2 <;> 

Proof state:
case h.e'_3.h.e'_6.h
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝¹ : HasColimit F
i : Discrete ι
inst✝ : HasStrictTerminalObjects C
U : Opens ↑↑(F.obj i).toPresheafedSpace
e : colimit.ι F i = colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv
H :
  OpenEmbedding ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).base
this :
  IsIso
    ((colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫ (preservesColimitIso forgetToPresheafedSpace F).inv).c.app
      (op
        ((IsOpenMap.functor
              (_ :
                IsOpenMap
                  ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                        (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
          U)))
e_1✝ : (Opens ↑↑(colimit F).toPresheafedSpace)ᵒᵖ = (Opens ↑↑(forgetToPresheafedSpace.obj (colimit F)))ᵒᵖ
⊢ op ((IsOpenMap.functor (_ : IsOpenMap ⇑(colimit.ι F i).base)).obj U) =
    op
      ((IsOpenMap.functor
            (_ :
              IsOpenMap
                ⇑(colimit.ι (F ⋙ forgetToPresheafedSpace) i ≫
                      (preservesColimitIso forgetToPresheafedSpace F).inv).base)).obj
        U)

Next tactic:
congr