Theorem cnvepresdmqs

Description: The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021)

	Ref	Expression
Assertion	cnvepresdmqs	⊢ ( ( ◡ E ↾ 𝐴 ) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-dmqs	⊢ ( ( ◡ E ↾ 𝐴 ) DomainQs 𝐴 ↔ ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 )
2		n0el3	⊢ ( ¬ ∅ ∈ 𝐴 ↔ ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 )
3	1 2	bitr4i	⊢ ( ( ◡ E ↾ 𝐴 ) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴 )