Binary Relation is Subclass of Product of Domain with Range

Theorem
Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.

Let $\Dom \RR$ denote the domain of $\RR$.

Let $\Img \RR$ denote the image of $\RR$.

Then:
 * $\RR$ is a subclass of $\Dom \RR \times \Img \RR$

Proof
Let $\tuple {x, y} \in \RR$.

Then by definition of domain of $\RR$:
 * $x \in \Dom \RR$

and by definition of image of $\RR$:
 * $y \in \Img \RR$

Hence by definition of Cartesian product:
 * $\tuple {x, y} \in \Dom \RR \times \Img \RR$

Hence the result by definition of subclass:
 * $\RR \subseteq \paren {\Dom \RR \times \Img \RR}$