Theorem petinidres

Description: A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres , disjALTVinidres and eqvrel1cossinidres . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petinidres	⊢ ( ( 𝑅 ∩ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ErALTV 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		petinidres2	⊢ ( ( Disj ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ∧ ( dom ( 𝑅 ∩ ( I ↾ 𝐴 ) ) / ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ) = 𝐴 ) ↔ ( EqvRel ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) / ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
2		dfpart2	⊢ ( ( 𝑅 ∩ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ( Disj ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ∧ ( dom ( 𝑅 ∩ ( I ↾ 𝐴 ) ) / ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
3		dferALTV2	⊢ ( ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ErALTV 𝐴 ↔ ( EqvRel ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) / ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( ( 𝑅 ∩ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ErALTV 𝐴 )