Theorem petidres

Description: A class is a partition by identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it, cf. eqvrel1cossidres . (Contributed by Peter Mazsa, 31-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		petidres2	$$⊢\left(Disj\left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\left({\mathrm{I}\upharpoonright }_A\right)/\left({\mathrm{I}\upharpoonright }_A\right)=A\right)\leftrightarrow \left(EqvRel\wr \left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr \left({\mathrm{I}\upharpoonright }_A\right)/\wr \left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$
2		dfpart2	Could not format ( ( _I |` A ) Part A <-> ( Disj ( _I |` A ) /\ ( dom ( _I |` A ) /. ( _I |` A ) ) = A ) ) : No typesetting found for |- ( ( _I |` A ) Part A <-> ( Disj ( _I |` A ) /\ ( dom ( _I |` A ) /. ( _I |` A ) ) = A ) ) with typecode |-
3		dferALTV2	$$⊢\wr \left({\mathrm{I}\upharpoonright }_A\right)ErALTVA\leftrightarrow \left(EqvRel\wr \left({\mathrm{I}\upharpoonright }_A\right)\wedge \mathrm{dom}\wr \left({\mathrm{I}\upharpoonright }_A\right)/\wr \left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$
4	1 2 3	3bitr4i	Could not format ( ( _I |` A ) Part A <-> ,~ ( _I |` A ) ErALTV A ) : No typesetting found for |- ( ( _I |` A ) Part A <-> ,~ ( _I |` A ) ErALTV A ) with typecode |-