Theorem elrn

Description: Membership in a range. (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Hypothesis	elrn.1	$$⊢A\in \mathrm{V}$$
Assertion	elrn	$$⊢A\in \mathrm{ran}B\leftrightarrow \exists xxBA$$

Proof

Step	Hyp	Ref	Expression
1		elrn.1	$$⊢A\in \mathrm{V}$$
2		elrng	$$⊢A\in \mathrm{V}\to \left(A\in \mathrm{ran}B\leftrightarrow \exists xxBA\right)$$
3	1 2	ax-mp	$$⊢A\in \mathrm{ran}B\leftrightarrow \exists xxBA$$