Theorem elima3

Description: Membership in an image. Theorem 34 of Suppes p. 65. (Contributed by NM, 14-Aug-1994)

	Ref	Expression
Hypothesis	elima.1	$$⊢A\in \mathrm{V}$$
Assertion	elima3	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\left(x\in C\wedge ⟨x,A⟩\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elima.1	$$⊢A\in \mathrm{V}$$
2	1	elima2	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\left(x\in C\wedge xBA\right)$$
3		df-br	$$⊢xBA\leftrightarrow ⟨x,A⟩\in B$$
4	3	anbi2i	$$⊢\left(x\in C\wedge xBA\right)\leftrightarrow \left(x\in C\wedge ⟨x,A⟩\in B\right)$$
5	4	exbii	$$⊢\exists x\left(x\in C\wedge xBA\right)\leftrightarrow \exists x\left(x\in C\wedge ⟨x,A⟩\in B\right)$$
6	2 5	bitri	$$⊢A\in B\left[C\right]\leftrightarrow \exists x\left(x\in C\wedge ⟨x,A⟩\in B\right)$$