Theorem: CategoryTheory.GrothendieckTopology.ext
File: Mathlib/CategoryTheory/Sites/Grothendieck.lean

tactic: cases J₁
state_before: C : Type u
inst✝ : Category.{v, u} C
X Y : C
S R : Sieve X
J J₁ J₂ : GrothendieckTopology C
h : J₁.sieves = J₂.sieves
⊢ J₁ = J₂
state_after: case mk
C : Type u
inst✝ : Category.{v, u} C
X Y : C
S R : Sieve X
J J₂ : GrothendieckTopology C
sieves✝ : (X : C) → Set (Sieve X)
top_mem'✝ : ∀ (X : C), ⊤ ∈ sieves✝ X
pullback_stable'✝ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝ X → Sieve.pullback f S ∈ sieves✝ Y
transitive'✝ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝ Y) → R ∈ sieves✝ X
h :
  { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝,
        transitive' := transitive'✝ }.sieves =
    J₂.sieves
⊢ { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝, transitive' := transitive'✝ } = J₂

tactic: cases J₂
state_before: case mk
C : Type u
inst✝ : Category.{v, u} C
X Y : C
S R : Sieve X
J J₂ : GrothendieckTopology C
sieves✝ : (X : C) → Set (Sieve X)
top_mem'✝ : ∀ (X : C), ⊤ ∈ sieves✝ X
pullback_stable'✝ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝ X → Sieve.pullback f S ∈ sieves✝ Y
transitive'✝ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝ Y) → R ∈ sieves✝ X
h :
  { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝,
        transitive' := transitive'✝ }.sieves =
    J₂.sieves
⊢ { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝, transitive' := transitive'✝ } = J₂
state_after: case mk.mk
C : Type u
inst✝ : Category.{v, u} C
X Y : C
S R : Sieve X
J : GrothendieckTopology C
sieves✝¹ : (X : C) → Set (Sieve X)
top_mem'✝¹ : ∀ (X : C), ⊤ ∈ sieves✝¹ X
pullback_stable'✝¹ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝¹ X → Sieve.pullback f S ∈ sieves✝¹ Y
transitive'✝¹ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝¹ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝¹ Y) → R ∈ sieves✝¹ X
sieves✝ : (X : C) → Set (Sieve X)
top_mem'✝ : ∀ (X : C), ⊤ ∈ sieves✝ X
pullback_stable'✝ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝ X → Sieve.pullback f S ∈ sieves✝ Y
transitive'✝ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝ Y) → R ∈ sieves✝ X
h :
  { sieves := sieves✝¹, top_mem' := top_mem'✝¹, pullback_stable' := pullback_stable'✝¹,
        transitive' := transitive'✝¹ }.sieves =
    { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝,
        transitive' := transitive'✝ }.sieves
⊢ { sieves := sieves✝¹, top_mem' := top_mem'✝¹, pullback_stable' := pullback_stable'✝¹, transitive' := transitive'✝¹ } =
    { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝, transitive' := transitive'✝ }

tactic: congr
state_before: case mk.mk
C : Type u
inst✝ : Category.{v, u} C
X Y : C
S R : Sieve X
J : GrothendieckTopology C
sieves✝¹ : (X : C) → Set (Sieve X)
top_mem'✝¹ : ∀ (X : C), ⊤ ∈ sieves✝¹ X
pullback_stable'✝¹ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝¹ X → Sieve.pullback f S ∈ sieves✝¹ Y
transitive'✝¹ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝¹ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝¹ Y) → R ∈ sieves✝¹ X
sieves✝ : (X : C) → Set (Sieve X)
top_mem'✝ : ∀ (X : C), ⊤ ∈ sieves✝ X
pullback_stable'✝ : ∀ ⦃X Y : C⦄ ⦃S : Sieve X⦄ (f : Y ⟶ X), S ∈ sieves✝ X → Sieve.pullback f S ∈ sieves✝ Y
transitive'✝ :
  ∀ ⦃X : C⦄ ⦃S : Sieve X⦄,
    S ∈ sieves✝ X →
      ∀ (R : Sieve X), (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ sieves✝ Y) → R ∈ sieves✝ X
h :
  { sieves := sieves✝¹, top_mem' := top_mem'✝¹, pullback_stable' := pullback_stable'✝¹,
        transitive' := transitive'✝¹ }.sieves =
    { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝,
        transitive' := transitive'✝ }.sieves
⊢ { sieves := sieves✝¹, top_mem' := top_mem'✝¹, pullback_stable' := pullback_stable'✝¹, transitive' := transitive'✝¹ } =
    { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝, transitive' := transitive'✝ }
state_after: no goals