Theorem erex

Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010) (Proof shortened by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Assertion	erex	$$⊢R\mathrm{Er}A\to \left(A\in V\to R\in \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		erssxp	$$⊢R\mathrm{Er}A\to R\subseteq A\times A$$
2		sqxpexg	$$⊢A\in V\to A\times A\in \mathrm{V}$$
3		ssexg	$$⊢\left(R\subseteq A\times A\wedge A\times A\in \mathrm{V}\right)\to R\in \mathrm{V}$$
4	1 2 3	syl2an	$$⊢\left(R\mathrm{Er}A\wedge A\in V\right)\to R\in \mathrm{V}$$
5	4	ex	$$⊢R\mathrm{Er}A\to \left(A\in V\to R\in \mathrm{V}\right)$$