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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 → ( dom 𝑅 / 𝑅 ) = ( dom ≀ 𝑅 / ≀ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		disjdmqsss	⊢ ( Disj 𝑅 → ( dom 𝑅 / 𝑅 ) ⊆ ( dom ≀ 𝑅 / ≀ 𝑅 ) )
2		disjdmqscossss	⊢ ( Disj 𝑅 → ( dom ≀ 𝑅 / ≀ 𝑅 ) ⊆ ( dom 𝑅 / 𝑅 ) )
3	1 2	eqssd	⊢ ( Disj 𝑅 → ( dom 𝑅 / 𝑅 ) = ( dom ≀ 𝑅 / ≀ 𝑅 ) )