Theorem mpet2

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet mpet3 , mostly in its conventional cpet and cpet2 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 with general R ). (Contributed by Peter Mazsa, 24-Sep-2021)

	Ref	Expression
Assertion	mpet2	$$⊢\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)PartA\leftrightarrow \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)ErALTVA$$

Proof

Step	Hyp	Ref	Expression
1		mpet	$$⊢MembPartA\leftrightarrow CoMembErA$$
2		df-membpart	$$⊢MembPartA\leftrightarrow \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)PartA$$
3		df-comember	$$⊢CoMembErA\leftrightarrow \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)ErALTVA$$
4	1 2 3	3bitr3i	$$⊢\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)PartA\leftrightarrow \wr \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)ErALTVA$$