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Metamath Proof Explorer

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 )