Theorem disjimeldisjdmqs

Description: Disj implies element-disjoint quotient carrier. Supplies the carrier-disjointness half of the Disjs pattern: under Disj R , the coset family is element-disjoint. (Contributed by Peter Mazsa, 5-Feb-2026)

	Ref	Expression
Assertion	disjimeldisjdmqs	$$⊢DisjR\to ElDisj\left(\mathrm{dom}R/R\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjim	$$⊢DisjR\to EqvRel\wr R$$
2		disjdmqs	$$⊢DisjR\to \mathrm{dom}R/R=\mathrm{dom}\wr R/\wr R$$
3	2	eqcomd	$$⊢DisjR\to \mathrm{dom}\wr R/\wr R=\mathrm{dom}R/R$$
4		eqvrelqseqdisj2	$$⊢\left(EqvRel\wr R\wedge \mathrm{dom}\wr R/\wr R=\mathrm{dom}R/R\right)\to ElDisj\left(\mathrm{dom}R/R\right)$$
5	1 3 4	syl2anc	$$⊢DisjR\to ElDisj\left(\mathrm{dom}R/R\right)$$