internlm/Lean-Github huggingface dataset:
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Tactic:
rw [Function.bijective_iff_existsUnique] at hFecs
Proof state after:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  ∀ (b : ↑{x | F.map (pullback.fst π π).op x = F.map (pullback.snd π π).op x}),
    ∃! a,
      MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a = b
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  ∀ (b : ↑{x | F.map (pullback.fst π π).op x = F.map (pullback.snd π π).op x}),
    ∃! a,
      MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a = b
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Tactic:
specialize hFecs ⟨(EqualizerFirstObjIso F π).hom y, iy_mem⟩
Proof state after:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
hFecs :
  ∃! a,
    MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a =
      { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem }
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
hFecs :
  ∃! a,
    MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a =
      { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem }
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Tactic:
obtain ⟨x, hx⟩ := hFecs
Proof state after:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (fun a =>
        MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
      x ∧
    ∀ (y_1 : F.obj B.op),
      (fun a =>
            MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π)
                a =
              { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
          y_1 →
        y_1 = x
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (fun a =>
        MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
      x ∧
    ∀ (y_1 : F.obj B.op),
      (fun a =>
            MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π)
                a =
              { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
          y_1 →
        y_1 = x
⊢ ∃! x, F.map π.op x = (EqualizerFirstObjIso F π).hom y
Tactic:
refine' ⟨x, _⟩
Proof state after:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (fun a =>
        MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π) a =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
      x ∧
    ∀ (y_1 : F.obj B.op),
      (fun a =>
            MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π)
                a =
              { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem })
          y_1 →
        y_1 = x
⊢ (fun x => F.map π.op x = (EqualizerFirstObjIso F π).hom y) x ∧
    ∀ (y_1 : F.obj B.op), (fun x => F.map π.op x = (EqualizerFirstObjIso F π).hom y) y_1 → y_1 = x
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  { val := F.map π.op x,
        property :=
          (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) x = (F.map π.op ≫ F.map (pullback.snd π π).op) x) } =
      { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } ∧
    ∀ (y_1 : F.obj B.op),
      { val := F.map π.op y_1,
            property :=
              (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) y_1 = (F.map π.op ≫ F.map (pullback.snd π π).op) y_1) } =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } →
        y_1 = x
⊢ ∀ (y_1 : F.obj B.op), F.map π.op y_1 = (EqualizerFirstObjIso F π).hom y → y_1 = x
Tactic:
intro z hz
Proof state after:
case uniq_F.intro
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  { val := F.map π.op x,
        property :=
          (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) x = (F.map π.op ≫ F.map (pullback.snd π π).op) x) } =
      { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } ∧
    ∀ (y_1 : F.obj B.op),
      { val := F.map π.op y_1,
            property :=
              (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) y_1 = (F.map π.op ≫ F.map (pullback.snd π π).op) y_1) } =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } →
        y_1 = x
z : F.obj B.op
hz : F.map π.op z = (EqualizerFirstObjIso F π).hom y
⊢ z = x
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case uniq_F.intro.a
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  { val := F.map π.op x,
        property :=
          (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) x = (F.map π.op ≫ F.map (pullback.snd π π).op) x) } =
      { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } ∧
    ∀ (y_1 : F.obj B.op),
      { val := F.map π.op y_1,
            property :=
              (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) y_1 = (F.map π.op ≫ F.map (pullback.snd π π).op) y_1) } =
          { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem } →
        y_1 = x
z : F.obj B.op
hz : F.map π.op z = (EqualizerFirstObjIso F π).hom y
⊢ { val := F.map π.op z,
      property := (_ : (F.map π.op ≫ F.map (pullback.fst π π).op) z = (F.map π.op ≫ F.map (pullback.snd π π).op) z) } =
    { val := (EqualizerFirstObjIso F π).hom y, property := iy_mem }
Tactic:
rwa [Subtype.ext_iff]
Proof state after:
no goals
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case inr.intro.intro.intro.intro.left
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) x = (EqualizerFirstObjIso F π).hom y ∧
    ∀ (y_1 : F.obj B.op),
      (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) y_1 =
          (EqualizerFirstObjIso F π).hom y →
        y_1 = x
⊢ (EqualizerFirstObjIso F π).hom (Equalizer.forkMap F (Presieve.singleton π) x) = (EqualizerFirstObjIso F π).hom y
Tactic:
exact hx.1
Proof state after:
no goals
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case inr.intro.intro.intro.intro.left.inj
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) x = (EqualizerFirstObjIso F π).hom y ∧
    ∀ (y_1 : F.obj B.op),
      (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) y_1 =
          (EqualizerFirstObjIso F π).hom y →
        y_1 = x
⊢ Function.Injective (EqualizerFirstObjIso F π).hom
Tactic:
apply Function.Bijective.injective
Proof state after:
case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) x = (EqualizerFirstObjIso F π).hom y ∧
    ∀ (y_1 : F.obj B.op),
      (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) y_1 =
          (EqualizerFirstObjIso F π).hom y →
        y_1 = x
⊢ Function.Bijective (EqualizerFirstObjIso F π).hom
Theorem: CompHaus.isSheafFor_of_Dagur
Proof state before:
case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) x = (EqualizerFirstObjIso F π).hom y ∧
    ∀ (y_1 : F.obj B.op),
      (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) y_1 =
          (EqualizerFirstObjIso F π).hom y →
        y_1 = x
⊢ Function.Bijective (EqualizerFirstObjIso F π).hom
Tactic:
rw [← isIso_iff_bijective]
Proof state after:
case inr.intro.intro.intro.intro.left.inj.hf
B : CompHaus
F : CompHausᵒᵖ ⥤ Type (u + 1)
hFpfp : PreservesFiniteProducts F
X : CompHaus
π : X ⟶ B
πsurj : Function.Surjective ↑π
y : Equalizer.FirstObj F (Presieve.singleton π)
h : Equalizer.Presieve.firstMap F (Presieve.singleton π) y = Equalizer.Presieve.secondMap F (Presieve.singleton π) y
hFecs :
  Function.Bijective
    (MapToEqualizer F π (pullback.fst π π) (pullback.snd π π) (_ : pullback.fst π π ≫ π = pullback.snd π π ≫ π))
fork_comp : Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom = F.map π.op
fmap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.fst π π).op =
    Equalizer.Presieve.firstMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
smap_comp :
  (EqualizerFirstObjIso F π).hom ≫ F.map (pullback.snd π π).op =
    Equalizer.Presieve.secondMap F (Presieve.singleton π) ≫ (EqualizerSecondObjIso F π).hom
iy_mem :
  F.map (pullback.fst π π).op ((EqualizerFirstObjIso F π).hom y) =
    F.map (pullback.snd π π).op ((EqualizerFirstObjIso F π).hom y)
x : F.obj B.op
hx :
  (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) x = (EqualizerFirstObjIso F π).hom y ∧
    ∀ (y_1 : F.obj B.op),
      (Equalizer.forkMap F (Presieve.singleton π) ≫ (EqualizerFirstObjIso F π).hom) y_1 =
          (EqualizerFirstObjIso F π).hom y →
        y_1 = x
⊢ IsIso (EqualizerFirstObjIso F π).hom