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		elrn2g	$$⊢A\in \mathrm{V}\to \left(A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B\right)$$
3	1 2	ax-mp	$$⊢A\in \mathrm{ran}B\leftrightarrow \exists x⟨x,A⟩\in B$$