Theorem brrelex12

Description: Two classes related by a binary relation are sets. (Contributed by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	brrelex12	$$⊢\left(\mathrm{Rel}R\wedge ARB\right)\to \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-rel	$$⊢\mathrm{Rel}R\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}$$
2	1	biimpi	$$⊢\mathrm{Rel}R\to R\subseteq \mathrm{V}\times \mathrm{V}$$
3	2	ssbrd	$$⊢\mathrm{Rel}R\to \left(ARB\to A\left(\mathrm{V}\times \mathrm{V}\right)B\right)$$
4	3	imp	$$⊢\left(\mathrm{Rel}R\wedge ARB\right)\to A\left(\mathrm{V}\times \mathrm{V}\right)B$$
5		brxp	$$⊢A\left(\mathrm{V}\times \mathrm{V}\right)B\leftrightarrow \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$
6	4 5	sylib	$$⊢\left(\mathrm{Rel}R\wedge ARB\right)\to \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$