Theorem mpets2

Description: Member Partition-Equivalence Theorem with binary relations, cf. mpet2 . (Contributed by Peter Mazsa, 24-Sep-2021)

	Ref	Expression
Assertion	mpets2	|- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) )

Proof

Step	Hyp	Ref	Expression
1		mpet2	|- ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A )
2		cnvepresex	|- ( A e. V -> ( `' _E |` A ) e. _V )
3		brpartspart	|- ( ( A e. V /\ ( `' _E |` A ) e. _V ) -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) )
4	2 3	mpdan	|- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) )
5		1cosscnvepresex	|- ( A e. V -> ,~ ( `' _E |` A ) e. _V )
6		brerser	|- ( ( A e. V /\ ,~ ( `' _E |` A ) e. _V ) -> ( ,~ ( `' _E |` A ) Ers A <-> ,~ ( `' _E |` A ) ErALTV A ) )
7	5 6	mpdan	|- ( A e. V -> ( ,~ ( `' _E |` A ) Ers A <-> ,~ ( `' _E |` A ) ErALTV A ) )
8	4 7	bibi12d	|- ( A e. V -> ( ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) <-> ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) ) )
9	1 8	mpbiri	|- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) )