Theorem elrelsrel

Description: The element of the relations class ( df-rels ) and the relation predicate are the same when R is a set. (Contributed by Peter Mazsa, 24-Nov-2018)

	Ref	Expression
Assertion	elrelsrel	$$⊢R\in V\to \left(R\in \mathrm{Rels}\leftrightarrow \mathrm{Rel}R\right)$$

Proof

Step	Hyp	Ref	Expression
1		elrels2	$$⊢R\in V\to \left(R\in \mathrm{Rels}\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}\right)$$
2		df-rel	$$⊢\mathrm{Rel}R\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}$$
3	1 2	bitr4di	$$⊢R\in V\to \left(R\in \mathrm{Rels}\leftrightarrow \mathrm{Rel}R\right)$$