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 /\ ( dom R /. R ) = A ) -> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		disjim	|- ( Disj R -> EqvRel ,~ R )
2	1	adantr	|- ( ( Disj R /\ ( dom R /. R ) = A ) -> EqvRel ,~ R )
3		disjdmqseq	|- ( Disj R -> ( ( dom R /. R ) = A <-> ( dom ,~ R /. ,~ R ) = A ) )
4	3	biimpa	|- ( ( Disj R /\ ( dom R /. R ) = A ) -> ( dom ,~ R /. ,~ R ) = A )
5	2 4	jca	|- ( ( Disj R /\ ( dom R /. R ) = A ) -> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) )