Theorem brdmqssqs

Description: If A and R are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021)

	Ref	Expression
Assertion	brdmqssqs	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		brdmqss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
2		df-dmqs	⊢ ( 𝑅 DomainQs 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 )
3	1 2	bitr4di	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴 ) )