Theorem elima2

Description: Membership in an image. Theorem 34 of Suppes p. 65. (Contributed by NM, 11-Aug-2004)

	Ref	Expression
Hypothesis	elima.1	$$⊢A\in \mathrm{V}$$
Assertion	elima2	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\left(x\in C\wedge xBA\right)$$

Proof

Step	Hyp	Ref	Expression
1		elima.1	$$⊢A\in \mathrm{V}$$
2	1	elima	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\in CxBA$$
3		df-rex	$$⊢\exists x\in CxBA\leftrightarrow \exists x\left(x\in C\wedge xBA\right)$$
4	2 3	bitri	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\left(x\in C\wedge xBA\right)$$