Theorem eldm

Description: Membership in a domain. Theorem 4 of Suppes p. 59. (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Hypothesis	eldm.1	⊢ 𝐴 ∈ V
Assertion	eldm	⊢ ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		eldm.1	⊢ 𝐴 ∈ V
2		eldmg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 𝐴 𝐵 𝑦 )