Theorem partim2

Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim . Lemma for petlem . (Contributed by Peter Mazsa, 17-Sep-2021)

	Ref	Expression
Assertion	partim2	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\to \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjim	$$⊢DisjR\to EqvRel\wr R$$
2	1	adantr	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\to EqvRel\wr R$$
3		disjdmqseq	$$⊢DisjR\to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}\wr R/\wr R=A\right)$$
4	3	biimpa	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\to \mathrm{dom}\wr R/\wr R=A$$
5	2 4	jca	$$⊢\left(DisjR\wedge \mathrm{dom}R/R=A\right)\to \left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=A\right)$$