Theorem petidres2

Description: Class A is a partition by the identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petidres2	$$⊢\left(Disj\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)/\left({\mathrm{I}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr \left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr \left({\mathrm{I}\upharpoonright }_A\right)/\wr \left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjALTVidres	$$⊢Disj\left({\mathrm{I}\upharpoonright }_A\right)$$
2	1	petlemi	$$⊢\left(Disj\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)/\left({\mathrm{I}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr \left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr \left({\mathrm{I}\upharpoonright }_A\right)/\wr \left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$