Theorem elrels2

Description: The element of the relations class ( df-rels ) and the relation predicate ( df-rel ) are the same when R is a set. (Contributed by Peter Mazsa, 14-Jun-2018)

	Ref	Expression
Assertion	elrels2	$$⊢R\in V\to \left(R\in \mathrm{Rels}\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-rels	$$⊢\mathrm{Rels}=𝒫\left(\mathrm{V}\times \mathrm{V}\right)$$
2	1	eleq2i	$$⊢R\in \mathrm{Rels}\leftrightarrow R\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)$$
3		elpwg	$$⊢R\in V\to \left(R\in 𝒫\left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}\right)$$
4	2 3	bitrid	$$⊢R\in V\to \left(R\in \mathrm{Rels}\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}\right)$$