Theorem petincnvepres

Description: The shortest form of a partition-equivalence theorem with intersection and general R . Cf. br1cossincnvepres . Cf. pet . (Contributed by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	petincnvepres	|- ( ( R i^i ( `' _E |` A ) ) Part A <-> ,~ ( R i^i ( `' _E |` A ) ) ErALTV A )

Proof

Step	Hyp	Ref	Expression
1		petincnvepres2	|- ( ( Disj ( R i^i ( `' _E |` A ) ) /\ ( dom ( R i^i ( `' _E |` A ) ) /. ( R i^i ( `' _E |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R i^i ( `' _E |` A ) ) /\ ( dom ,~ ( R i^i ( `' _E |` A ) ) /. ,~ ( R i^i ( `' _E |` A ) ) ) = A ) )
2		dfpart2	|- ( ( R i^i ( `' _E |` A ) ) Part A <-> ( Disj ( R i^i ( `' _E |` A ) ) /\ ( dom ( R i^i ( `' _E |` A ) ) /. ( R i^i ( `' _E |` A ) ) ) = A ) )
3		dferALTV2	|- ( ,~ ( R i^i ( `' _E |` A ) ) ErALTV A <-> ( EqvRel ,~ ( R i^i ( `' _E |` A ) ) /\ ( dom ,~ ( R i^i ( `' _E |` A ) ) /. ,~ ( R i^i ( `' _E |` A ) ) ) = A ) )
4	1 2 3	3bitr4i	|- ( ( R i^i ( `' _E |` A ) ) Part A <-> ,~ ( R i^i ( `' _E |` A ) ) ErALTV A )