Theorem rnexg

Description: The range of a set is a set. Corollary 6.8(3) of TakeutiZaring p. 26. Similar to Lemma 3D of Enderton p. 41. (Contributed by NM, 31-Mar-1995)

	Ref	Expression
Assertion	rnexg	$$⊢A\in V\to \mathrm{ran}A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		uniexg	$$⊢A\in V\to \bigcup A\in \mathrm{V}$$
2		uniexg	$$⊢\bigcup A\in \mathrm{V}\to \bigcup \bigcup A\in \mathrm{V}$$
3		ssun2	$$⊢\mathrm{ran}A\subseteq \mathrm{dom}A\cup \mathrm{ran}A$$
4		dmrnssfld	$$⊢\mathrm{dom}A\cup \mathrm{ran}A\subseteq \bigcup \bigcup A$$
5	3 4	sstri	$$⊢\mathrm{ran}A\subseteq \bigcup \bigcup A$$
6		ssexg	$$⊢\left(\mathrm{ran}A\subseteq \bigcup \bigcup A\wedge \bigcup \bigcup A\in \mathrm{V}\right)\to \mathrm{ran}A\in \mathrm{V}$$
7	5 6	mpan	$$⊢\bigcup \bigcup A\in \mathrm{V}\to \mathrm{ran}A\in \mathrm{V}$$
8	1 2 7	3syl	$$⊢A\in V\to \mathrm{ran}A\in \mathrm{V}$$