Theorem petidres

Description: A class is a partition by identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it, cf. eqvrel1cossidres . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petidres	⊢ ( ( I ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( I ↾ 𝐴 ) ErALTV 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		petidres2	⊢ ( ( Disj ( I ↾ 𝐴 ) ∧ ( dom ( I ↾ 𝐴 ) / ( I ↾ 𝐴 ) ) = 𝐴 ) ↔ ( EqvRel ≀ ( I ↾ 𝐴 ) ∧ ( dom ≀ ( I ↾ 𝐴 ) / ≀ ( I ↾ 𝐴 ) ) = 𝐴 ) )
2		dfpart2	⊢ ( ( I ↾ 𝐴 ) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴 ) ∧ ( dom ( I ↾ 𝐴 ) / ( I ↾ 𝐴 ) ) = 𝐴 ) )
3		dferALTV2	⊢ ( ≀ ( I ↾ 𝐴 ) ErALTV 𝐴 ↔ ( EqvRel ≀ ( I ↾ 𝐴 ) ∧ ( dom ≀ ( I ↾ 𝐴 ) / ≀ ( I ↾ 𝐴 ) ) = 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( ( I ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( I ↾ 𝐴 ) ErALTV 𝐴 )