Theorem petid

Description: A class 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	petid	⊢ ( I Part 𝐴 ↔ ≀ I ErALTV 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		petid2	⊢ ( ( Disj I ∧ ( dom I / I ) = 𝐴 ) ↔ ( EqvRel ≀ I ∧ ( dom ≀ I / ≀ I ) = 𝐴 ) )
2		dfpart2	⊢ ( I Part 𝐴 ↔ ( Disj I ∧ ( dom I / I ) = 𝐴 ) )
3		dferALTV2	⊢ ( ≀ I ErALTV 𝐴 ↔ ( EqvRel ≀ I ∧ ( dom ≀ I / ≀ I ) = 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( I Part 𝐴 ↔ ≀ I ErALTV 𝐴 )