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Metamath Proof Explorer

Theorem petidres2

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

		Ref	Expression
	Assertion	petidres2	⊢ ( ( Disj ( I ↾ 𝐴 ) ∧ ( dom ( I ↾ 𝐴 ) / ( I ↾ 𝐴 ) ) = 𝐴 ) ↔ ( EqvRel ≀ ( I ↾ 𝐴 ) ∧ ( dom ≀ ( I ↾ 𝐴 ) / ≀ ( I ↾ 𝐴 ) ) = 𝐴 ) )

Proof

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