Theorem petincnvepres

Description: The shortest form of a partition-equivalence theorem with intersection and general R . Cf. br1cossincnvepres . Cf. pet . (Contributed by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	petincnvepres	$$⊢\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)PartA\leftrightarrow \wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)ErALTVA$$

Proof

Step	Hyp	Ref	Expression
1		petincnvepres2	$$⊢\left(Disj\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)/\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)=A\right)\leftrightarrow \left(EqvRel\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)/\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)=A\right)$$
2		dfpart2	$$⊢\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)PartA\leftrightarrow \left(Disj\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)/\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)=A\right)$$
3		dferALTV2	$$⊢\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)ErALTVA\leftrightarrow \left(EqvRel\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)\wedge \mathrm{dom}\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)/\wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)=A\right)$$
4	1 2 3	3bitr4i	$$⊢\left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)PartA\leftrightarrow \wr \left(R\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)ErALTVA$$