Theorem elrn2g

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	elrn2g	$$⊢A\in V\to \left(A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		opeq2	$$⊢y=A\to ⟨x,y⟩=⟨x,A⟩$$
2	1	eleq1d	$$⊢y=A\to \left(⟨x,y⟩\in B\leftrightarrow ⟨x,A⟩\in B\right)$$
3	2	exbidv	$$⊢y=A\to \left(\exists x⟨x,y⟩\in B\leftrightarrow \exists x⟨x,A⟩\in B\right)$$
4		dfrn3	$$⊢\mathrm{ran}B=\left\{y|\exists x⟨x,y⟩\in B\right\}$$
5	3 4	elab2g	$$⊢A\in V\to \left(A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B\right)$$