Theorem brdmqss

Description: The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019)

	Ref	Expression
Assertion	brdmqss	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dmqseq	⊢ ( 𝑥 = 𝑅 → ( dom 𝑥 / 𝑥 ) = ( dom 𝑅 / 𝑅 ) )
2		id	⊢ ( 𝑦 = 𝐴 → 𝑦 = 𝐴 )
3	1 2	eqeqan12d	⊢ ( ( 𝑥 = 𝑅 ∧ 𝑦 = 𝐴 ) → ( ( dom 𝑥 / 𝑥 ) = 𝑦 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
4		df-dmqss	⊢ DomainQss = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( dom 𝑥 / 𝑥 ) = 𝑦 }
5	3 4	brabga	⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → ( 𝑅 DomainQss 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
6	5	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 DomainQss 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )