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^i ( _I |` A ) ) Part A <-> ,~ ( R i^i ( _I |` A ) ) ErALTV A )

Proof

Step	Hyp	Ref	Expression
1		petinidres2	|- ( ( Disj ( R i^i ( _I |` A ) ) /\ ( dom ( R i^i ( _I |` A ) ) /. ( R i^i ( _I |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R i^i ( _I |` A ) ) /\ ( dom ,~ ( R i^i ( _I |` A ) ) /. ,~ ( R i^i ( _I |` A ) ) ) = A ) )
2		dfpart2	|- ( ( R i^i ( _I |` A ) ) Part A <-> ( Disj ( R i^i ( _I |` A ) ) /\ ( dom ( R i^i ( _I |` A ) ) /. ( R i^i ( _I |` A ) ) ) = A ) )
3		dferALTV2	|- ( ,~ ( R i^i ( _I |` A ) ) ErALTV A <-> ( EqvRel ,~ ( R i^i ( _I |` A ) ) /\ ( dom ,~ ( R i^i ( _I |` A ) ) /. ,~ ( R i^i ( _I |` A ) ) ) = A ) )
4	1 2 3	3bitr4i	|- ( ( R i^i ( _I |` A ) ) Part A <-> ,~ ( R i^i ( _I |` A ) ) ErALTV A )