Theorem disjdmqs

Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 and petlem via disjdmqseq . (Contributed by Peter Mazsa, 16-Sep-2021)

	Ref	Expression
Assertion	disjdmqs	$$⊢DisjR\to \mathrm{dom}R/R=\mathrm{dom}\wr R/\wr R$$

Proof

Step	Hyp	Ref	Expression
1		disjdmqsss	$$⊢DisjR\to \mathrm{dom}R/R\subseteq \mathrm{dom}\wr R/\wr R$$
2		disjdmqscossss	$$⊢DisjR\to \mathrm{dom}\wr R/\wr R\subseteq \mathrm{dom}R/R$$
3	1 2	eqssd	$$⊢DisjR\to \mathrm{dom}R/R=\mathrm{dom}\wr R/\wr R$$