Theorem eldm2

Description: Membership in a domain. Theorem 4 of Suppes p. 59. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Hypothesis	eldm.1	⊢ 𝐴 ∈ V
Assertion	eldm2	⊢ ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eldm.1	⊢ 𝐴 ∈ V
2		eldm2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ dom 𝐵 ↔ ∃ 𝑦 ⟨ 𝐴 , 𝑦 ⟩ ∈ 𝐵 )