Theorem pet2

Description: Partition-Equivalence Theorem, with general R . This theorem (together with pet and pets ) is the main result of my investigation into set theory, see the comment of pet . (Contributed by Peter Mazsa, 24-May-2021) (Revised by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	pet2	$$⊢\left(DisjR⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\wedge \mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj5	$$⊢\left(EqvRel\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\right)\to DisjR⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
2	1	petlem	$$⊢\left(DisjR⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\wedge \mathrm{dom}R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)/\wr R⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)=A\right)$$