Theorem eldm2

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

	Ref	Expression
Hypothesis	eldm.1	$$⊢A\in \mathrm{V}$$
Assertion	eldm2	$$⊢A\in \mathrm{dom}B\leftrightarrow \exists y⟨A,y⟩\in B$$

Proof

Step	Hyp	Ref	Expression
1		eldm.1	$$⊢A\in \mathrm{V}$$
2		eldm2g	$$⊢A\in \mathrm{V}\to \left(A\in \mathrm{dom}B\leftrightarrow \exists y⟨A,y⟩\in B\right)$$
3	1 2	ax-mp	$$⊢A\in \mathrm{dom}B\leftrightarrow \exists y⟨A,y⟩\in B$$