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Metamath Proof Explorer

Theorem partim2

Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim . Lemma for petlem . (Contributed by Peter Mazsa, 17-Sep-2021)

		Ref	Expression
	Assertion	partim2	$⊢ (Disj R \land dom R / R = A) \to (EqvRel ≀ R \land dom ≀ R / ≀ R = A)$

Proof

Step	Hyp	Ref	Expression
1		disjim	$⊢ Disj R \to EqvRel ≀ R$
2	1	adantr	$⊢ (Disj R \land dom R / R = A) \to EqvRel ≀ R$
3		disjdmqseq	$⊢ Disj R \to (dom R / R = A \leftrightarrow dom ≀ R / ≀ R = A)$
4	3	biimpa	$⊢ (Disj R \land dom R / R = A) \to dom ≀ R / ≀ R = A$
5	2 4	jca	$⊢ (Disj R \land dom R / R = A) \to (EqvRel ≀ R \land dom ≀ R / ≀ R = A)$