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:

$\RR \subseteq \paren {\Dom \RR \times \Img \RR}$

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations