Theorem elrn2

Description: Membership in a range. (Contributed by NM, 10-Jul-1994)

	Ref	Expression
Hypothesis	elrn.1	$$⊢A\in \mathrm{V}$$
Assertion	elrn2	$$⊢A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B$$

Proof

Step	Hyp	Ref	Expression
1		elrn.1	$$⊢A\in \mathrm{V}$$
2		opeq2	$$⊢y=A\to ⟨x,y⟩=⟨x,A⟩$$
3	2	eleq1d	$$⊢y=A\to \left(⟨x,y⟩\in B\leftrightarrow ⟨x,A⟩\in B\right)$$
4	3	exbidv	$$⊢y=A\to \left(\exists x⟨x,y⟩\in B\leftrightarrow \exists x⟨x,A⟩\in B\right)$$
5		dfrn3	$$⊢\mathrm{ran}B=\left\{y|\exists x⟨x,y⟩\in B\right\}$$
6	1 4 5	elab2	$$⊢A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B$$