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 ) )