Theorem eldm

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

	Ref	Expression
Hypothesis	eldm.1	$$⊢A\in \mathrm{V}$$
Assertion	eldm	$$⊢A\in \mathrm{dom}B\leftrightarrow \exists yABy$$

Proof

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