equalizerSieve_eq_top_iff — Mathlib

English

For any F: C^op ⥤ D and X in C^op, the equalizer sieve of x against itself equals the top sieve on X.

Русский

Для произвольного F: C^op ⥤ D и X, равнитель Sieve для x против самого себя равен верхней сиве на X.

LaTeX

$$$\forall F:X\mapsto D,\ equalizerSieve x x = \top$$$

Lean4

@[simp]
theorem equalizerSieve_eq_top_iff {F : Cᵒᵖ ⥤ D} {X : Cᵒᵖ} (x y : ToType (F.obj X)) : equalizerSieve x y = ⊤ ↔ x = y :=
  by
  constructor
  · intro h
    simpa using (show equalizerSieve x y (𝟙 _) by simp [h])
  · rintro rfl
    apply equalizerSieve_self_eq_top

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