Theorem dmqseqim2

Description: Lemma for erim2 . (Contributed by Peter Mazsa, 29-Dec-2021)

	Ref	Expression
Assertion	dmqseqim2	⊢ ( 𝑅 ∈ 𝑉 → ( Rel 𝑅 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 → ( 𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dmqseqim	⊢ ( 𝑅 ∈ 𝑉 → ( Rel 𝑅 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 → ran 𝑅 = ∪ 𝐴 ) ) )
2		eleq2	⊢ ( ran 𝑅 = ∪ 𝐴 → ( 𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴 ) )
3	1 2	syl8	⊢ ( 𝑅 ∈ 𝑉 → ( Rel 𝑅 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 → ( 𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴 ) ) ) )