Theorem imaexg

Description: The image of a set is a set. Theorem 3.17 of Monk1 p. 39. (Contributed by NM, 24-Jul-1995)

	Ref	Expression
Assertion	imaexg	$$⊢A\in V\to A\left[B\right]\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		imassrn	$$⊢A\left[B\right]\subseteq \mathrm{ran}A$$
2		rnexg	$$⊢A\in V\to \mathrm{ran}A\in \mathrm{V}$$
3		ssexg	$$⊢\left(A\left[B\right]\subseteq \mathrm{ran}A\wedge \mathrm{ran}A\in \mathrm{V}\right)\to A\left[B\right]\in \mathrm{V}$$
4	1 2 3	sylancr	$$⊢A\in V\to A\left[B\right]\in \mathrm{V}$$