Metamath Proof Explorer

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 syl6bbr ⊢ A ∈ V ∧ R ∈ W → R DomainQss A ↔ R DomainQs A