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Metamath Proof Explorer

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	$⊢ Disj R \to dom R / R = dom ≀ R / ≀ R$

Proof

Step	Hyp	Ref	Expression
1		disjdmqsss	$⊢ Disj R \to dom R / R \subseteq dom ≀ R / ≀ R$
2		disjdmqscossss	$⊢ Disj R \to dom ≀ R / ≀ R \subseteq dom R / R$
3	1 2	eqssd	$⊢ Disj R \to dom R / R = dom ≀ R / ≀ R$