This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem petid2

Description: Class A is a partition by the identity class if and only if the cosets by the identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021)

		Ref	Expression
	Assertion	petid2	⊢ ( ( Disj I ∧ ( dom I / I ) = 𝐴 ) ↔ ( EqvRel ≀ I ∧ ( dom ≀ I / ≀ I ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		disjALTVid	⊢ Disj I
2	1	petlemi	⊢ ( ( Disj I ∧ ( dom I / I ) = 𝐴 ) ↔ ( EqvRel ≀ I ∧ ( dom ≀ I / ≀ I ) = 𝐴 ) )