Theorem unidmqseq

Description: The union of the domain quotient of a relation is equal to the class A if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019) (Revised by Peter Mazsa, 28-Dec-2021)

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

Proof

Step	Hyp	Ref	Expression
1		unidmqs	⊢ ( 𝑅 ∈ 𝑉 → ( Rel 𝑅 → ∪ ( dom 𝑅 / 𝑅 ) = ran 𝑅 ) )
2	1	imp	⊢ ( ( 𝑅 ∈ 𝑉 ∧ Rel 𝑅 ) → ∪ ( dom 𝑅 / 𝑅 ) = ran 𝑅 )
3	2	eqeq1d	⊢ ( ( 𝑅 ∈ 𝑉 ∧ Rel 𝑅 ) → ( ∪ ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ran 𝑅 = 𝐴 ) )
4	3	ex	⊢ ( 𝑅 ∈ 𝑉 → ( Rel 𝑅 → ( ∪ ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ran 𝑅 = 𝐴 ) ) )