Theorem partim

Description: Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 . (Contributed by Peter Mazsa, 17-Sep-2021)

	Ref	Expression
Assertion	partim	$$⊢RPartA\to \wr RErALTVA$$

Proof

Step	Hyp	Ref	Expression
1		partim2	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\to \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$
2		dfpart2	$$⊢RPartA\leftrightarrow \left(DisjR\wedge \mathrm{dom}R/R=A\right)$$
3		dferALTV2	$$⊢\wr RErALTVA\leftrightarrow \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$
4	1 2 3	3imtr4i	$$⊢RPartA\to \wr RErALTVA$$