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	|- ( Disj R -> ( dom R /. R ) = ( dom ,~ R /. ,~ R ) )

Proof

Step	Hyp	Ref	Expression
1		disjdmqsss	|- ( Disj R -> ( dom R /. R ) C_ ( dom ,~ R /. ,~ R ) )
2		disjdmqscossss	|- ( Disj R -> ( dom ,~ R /. ,~ R ) C_ ( dom R /. R ) )
3	1 2	eqssd	|- ( Disj R -> ( dom R /. R ) = ( dom ,~ R /. ,~ R ) )