-- Mathlib/CategoryTheory/Sites/Equivalence.lean
lemma W_inverseImage_whiskeringLeft :
    K.W.inverseImage ((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op) = J.W

Proof state:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ K.W = LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj

Tactic:
rw [W_eq_W_range_sheafToPresheaf_obj, ← LeftBousfield.W_isoClosure]

Result:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ LeftBousfield.W (ObjectProperty.isoClosure fun x => x ∈ Set.range (sheafToPresheaf K A).obj) =
    LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj

Proof state:
case h
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
h₁ : K.W = LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
h₂ : ∀ (R : Sheaf J A), (Function.Bijective fun g => whiskerLeft G.op f ≫ g) ↔ Function.Bijective fun g => f ≫ g
⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj)
      (((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).map f) ↔
    LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) f

Tactic:
constructor

Result:
case h.mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
h₁ : K.W = LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
h₂ : ∀ (R : Sheaf J A), (Function.Bijective fun g => whiskerLeft G.op f ≫ g) ↔ Function.Bijective fun g => f ≫ g
⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj)
      (((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).map f) →
    LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) f

case h.mpr
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
h₁ : K.W = LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
h₂ : ∀ (R : Sheaf J A), (Function.Bijective fun g => whiskerLeft G.op f ≫ g) ↔ Function.Bijective fun g => f ≫ g
⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) f →
    LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj)
      (((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).map f)

Proof state:
case h.h.a.mpr.intro.intro.intro.intro
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
R : Sheaf J A
e : P ≅ (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj R
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf K A).obj) P

Tactic:
exact ⟨G.op ⋙ R.val, ⟨(G.sheafPushforwardContinuous A K J).obj R, rfl⟩, ⟨e⟩⟩

Result:
no goals

Proof state:
case h.h.a
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf K A).obj) P ↔
    ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj) P

Tactic:
constructor

Result:
case h.h.a.mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf K A).obj) P →
    ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj) P

case h.h.a.mpr
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj) P →
    ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf K A).obj) P

Proof state:
case h.h.a.mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf K A).obj) P →
    ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj) P

Tactic:
rintro ⟨_, ⟨R, rfl⟩, ⟨e⟩⟩

Result:
case h.h.a.mp.intro.intro.intro.intro
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P✝ Q : Cᵒᵖ ⥤ A
f : P✝ ⟶ Q
P : Dᵒᵖ ⥤ A
R : Sheaf K A
e : P ≅ (sheafToPresheaf K A).obj R
⊢ ObjectProperty.isoClosure (fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj) P