-- Mathlib/CategoryTheory/Sites/Closed.lean
theorem topology_eq_iff_same_sheaves {J₁ J₂ : GrothendieckTopology C} :
    J₁ = J₂ ↔ ∀ P : Cᵒᵖ ⥤ Type max v u, Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P

Proof state:
case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ = J₂

Tactic:
apply le_antisymm

Result:
case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ ≤ J₂

case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₂ ≤ J₁

Proof state:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₁)

Tactic:
rw [← h]

Result:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₁)

Proof state:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₂)

Tactic:
apply classifier_isSheaf

Result:
no goals

Proof state:
case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ ≤ J₂

Tactic:
apply le_topology_of_closedSieves_isSheaf

Result:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₂)

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
⊢ J₁ = J₂ → ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P

Tactic:
rintro rfl

Result:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
⊢ ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₁ P

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type (max v u)
⊢ Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₁ P

Tactic:
rfl

Result:
no goals

Proof state:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
⊢ J₁ = J₂ ↔ ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P

Tactic:
constructor

Result:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
⊢ J₁ = J₂ → ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P

case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
⊢ (∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P) → J₁ = J₂

Proof state:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₂)

Tactic:
rw [h]

Result:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₂)

Proof state:
case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
⊢ (∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P) → J₁ = J₂

Tactic:
intro h

Result:
case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ = J₂

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
⊢ ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₁ P

Tactic:
intro P

Result:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type (max v u)
⊢ Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₁ P

Proof state:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₁)

Tactic:
apply classifier_isSheaf

Result:
no goals

Proof state:
case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₂ ≤ J₁

Tactic:
apply le_topology_of_closedSieves_isSheaf

Result:
case mpr.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₁)