Theorem mpets2

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

	Ref	Expression
Assertion	mpets2	Could not format assertion : No typesetting found for |- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		mpet2	Could not format ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) : No typesetting found for |- ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) with typecode |-
2		cnvepresex	$$⊢A\in V\to {{\mathrm{E}}^{-1}\upharpoonright }_A\in \mathrm{V}$$
3		brpartspart	Could not format ( ( A e. V /\ ( `' _E |` A ) e. _V ) -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) ) : No typesetting found for |- ( ( A e. V /\ ( `' _E |` A ) e. _V ) -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) ) with typecode |-
4	2 3	mpdan	Could not format ( A e. V -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) ) : No typesetting found for |- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) ) with typecode |-
5		1cosscnvepresex	$$⊢A\in V\to \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\in \mathrm{V}$$
6		brerser	$$⊢\left(A\in V\wedge \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\in \mathrm{V}\right)\to \left(\wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\mathrm{Ers}A\leftrightarrow \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)ErALTVA\right)$$
7	5 6	mpdan	$$⊢A\in V\to \left(\wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\mathrm{Ers}A\leftrightarrow \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)ErALTVA\right)$$
8	4 7	bibi12d	Could not format ( A e. V -> ( ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) <-> ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) ) ) : No typesetting found for |- ( A e. V -> ( ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) <-> ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) ) ) with typecode |-
9	1 8	mpbiri	Could not format ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) ) : No typesetting found for |- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) ) with typecode |-