Theorem petinidres

Description: A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres , disjALTVinidres and eqvrel1cossinidres . (Contributed by Peter Mazsa, 31-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		petinidres2	$$⊢\left(Disj\left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)/\left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)=A\right)\leftrightarrow \left(EqvRel\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)/\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)=A\right)$$
2		dfpart2	Could not format ( ( R i^i ( _I |` A ) ) Part A <-> ( Disj ( R i^i ( _I |` A ) ) /\ ( dom ( R i^i ( _I |` A ) ) /. ( R i^i ( _I |` A ) ) ) = A ) ) : No typesetting found for |- ( ( R i^i ( _I |` A ) ) Part A <-> ( Disj ( R i^i ( _I |` A ) ) /\ ( dom ( R i^i ( _I |` A ) ) /. ( R i^i ( _I |` A ) ) ) = A ) ) with typecode |-
3		dferALTV2	$$⊢\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)ErALTVA\leftrightarrow \left(EqvRel\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)/\wr \left(R\cap \left({\mathrm{I}\upharpoonright }_A\right)\right)=A\right)$$
4	1 2 3	3bitr4i	Could not format ( ( R i^i ( _I |` A ) ) Part A <-> ,~ ( R i^i ( _I |` A ) ) ErALTV A ) : No typesetting found for |- ( ( R i^i ( _I |` A ) ) Part A <-> ,~ ( R i^i ( _I |` A ) ) ErALTV A ) with typecode |-