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Theorems · Theorem · order theory

Setoid.ker_eq_lift_of_injective

∀ {α : Type u_1} {β : Type u_2} {r : Setoid α} (f : α → β) (H : r ≤ Setoid.ker f),
  Function.Injective (Quotient.lift f H) → Setoid.ker f = r

Given a map f from α to β, the kernel of f is the unique equivalence relation on α whose induced map from the quotient of α to β is injective.

Defined in
Mathlib.Data.Setoid.Basic
Cited by
4 results in Mathlib
Foundations
Depth 12 from the axioms · uses propext, Quot.sound
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Around this declaration

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RingCon.lift_injective_iff · cited by 1RingCon.lift_injective_iffCon.ker_eq_lift_of_injective · cited by 0Con.ker_eq_lift_of_inject…Setoid.lift_injective_iff_ker_eq_of_le · cited by 0Setoid.lift_injective_iff…AddCon.ker_eq_lift_of_injective · cited by 0AddCon.ker_eq_lift_of_inj…le_antisymm · cited by 2068le_antisymmSetoid.ker · cited by 43Setoid.kerSetoid.ker_eq_lift_of_injecti…CITED BYCITES

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