Theorem petxrnidres

Description: A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres , disjALTVxrnidres and eqvrel1cossxrnidres . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petxrnidres	|- ( ( R |X. ( _I |` A ) ) Part A <-> ,~ ( R |X. ( _I |` A ) ) ErALTV A )

Proof

Step	Hyp	Ref	Expression
1		petxrnidres2	|- ( ( Disj ( R |X. ( _I |` A ) ) /\ ( dom ( R |X. ( _I |` A ) ) /. ( R |X. ( _I |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R |X. ( _I |` A ) ) /\ ( dom ,~ ( R |X. ( _I |` A ) ) /. ,~ ( R |X. ( _I |` A ) ) ) = A ) )
2		dfpart2	|- ( ( R |X. ( _I |` A ) ) Part A <-> ( Disj ( R |X. ( _I |` A ) ) /\ ( dom ( R |X. ( _I |` A ) ) /. ( R |X. ( _I |` A ) ) ) = A ) )
3		dferALTV2	|- ( ,~ ( R |X. ( _I |` A ) ) ErALTV A <-> ( EqvRel ,~ ( R |X. ( _I |` A ) ) /\ ( dom ,~ ( R |X. ( _I |` A ) ) /. ,~ ( R |X. ( _I |` A ) ) ) = A ) )
4	1 2 3	3bitr4i	|- ( ( R |X. ( _I |` A ) ) Part A <-> ,~ ( R |X. ( _I |` A ) ) ErALTV A )