-- Mathlib/AlgebraicGeometry/Morphisms/Separated.lean
lemma Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective
    (hf : Function.Injective f.base) :
    diagonalCoverDiagonalRange f 𝒰 𝒱 = ⊤

Proof state:
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
⊢ x ∈ ↑(diagonalCoverDiagonalRange f 𝒰 𝒱)

Tactic:
simp only [diagonalCoverDiagonalRange, openCoverOfBase_J, openCoverOfBase_obj,
  openCoverOfLeftRight_J, Opens.iSup_mk, Opens.carrier_eq_coe, Hom.coe_opensRange, Opens.coe_mk,
  Set.mem_iUnion, Set.mem_range, Sigma.exists]

Result:
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Proof state:
case intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Tactic:
obtain ⟨z, hz₁, hz₂⟩ := exists_preimage_pullback _ _ hy.symm

Result:
case intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
⊢ ∃ a b y, (ConcreteCategory.hom ((diagonalCover f 𝒰 𝒱).map ⟨a, (b, b)⟩).base) y = x

Proof state:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ x ∈
    Set.range
      ⇑(ConcreteCategory.hom
          (pullback.map ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) f f
              ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯).base)

Tactic:
rw [range_map]

Result:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ x ∈
    ⇑(ConcreteCategory.hom (pullback.fst f f).base) ⁻¹'
        Set.range ⇑(ConcreteCategory.hom ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)).base) ∩
      ⇑(ConcreteCategory.hom (pullback.snd f f).base) ⁻¹'
        Set.range ⇑(ConcreteCategory.hom ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)).base)

Proof state:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ ∃ y,
    (ConcreteCategory.hom
          (pullback.map ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) f f
              ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯).base)
        y =
      x

Tactic:
show x ∈ Set.range _

Result:
case intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
hf : Function.Injective ⇑(ConcreteCategory.hom f.base)
x : ↑↑(pullback.diagonalObj f).toPresheafedSpace
H : (ConcreteCategory.hom (pullback.fst f f).base) x = (ConcreteCategory.hom (pullback.snd f f).base) x
i : 𝒰.J := 𝒰.f ((ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x))
y : ↑↑(𝒰.obj i).toPresheafedSpace
hy✝ :
  (ConcreteCategory.hom (𝒰.map i).base) y =
    (ConcreteCategory.hom f.base) ((ConcreteCategory.hom (pullback.fst f f).base) x)
z : ↑↑(pullback f (𝒰.map i)).toPresheafedSpace
hz₁ : (ConcreteCategory.hom (pullback.fst f (𝒰.map i)).base) z = (ConcreteCategory.hom (pullback.fst f f).base) x
hz₂ : (ConcreteCategory.hom (pullback.snd f (𝒰.map i)).base) z = y
j : (𝒱 i).J := (𝒱 i).f z
w : ↑↑((𝒱 i).obj j).toPresheafedSpace
hy : (ConcreteCategory.hom ((𝒱 i).map j).base) w = z
⊢ x ∈
    Set.range
      ⇑(ConcreteCategory.hom
          (pullback.map ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) ((𝒱 i).map j ≫ Cover.pullbackHom 𝒰 f i) f f
              ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒱 i).map j ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯).base)