Theorem petxrnidres

Description: A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres , disjALTVxrnidres and eqvrel1cossxrnidres . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petxrnidres	Could not format assertion : No typesetting found for |- ( ( R |X. ( _I |` A ) ) Part A <-> ,~ ( R |X. ( _I |` A ) ) ErALTV A ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		petxrnidres2	$$⊢\left(DisjR⋉\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}R⋉\left({\mathrm{I}\upharpoonright }_A\right)/R⋉\left({\mathrm{I}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)/\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$
2		dfpart2	Could not format ( ( R |X. ( _I |` A ) ) Part A <-> ( Disj ( R |X. ( _I |` A ) ) /\ ( dom ( R |X. ( _I |` A ) ) /. ( R |X. ( _I |` A ) ) ) = A ) ) : No typesetting found for |- ( ( R |X. ( _I |` A ) ) Part A <-> ( Disj ( R |X. ( _I |` A ) ) /\ ( dom ( R |X. ( _I |` A ) ) /. ( R |X. ( _I |` A ) ) ) = A ) ) with typecode |-
3		dferALTV2	$$⊢\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)ErALTVA\leftrightarrow \left(EqvRel\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)/\wr R⋉\left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$
4	1 2 3	3bitr4i	Could not format ( ( R |X. ( _I |` A ) ) Part A <-> ,~ ( R |X. ( _I |` A ) ) ErALTV A ) : No typesetting found for |- ( ( R |X. ( _I |` A ) ) Part A <-> ,~ ( R |X. ( _I |` A ) ) ErALTV A ) with typecode |-