Relation is Set implies Domain and Image are Sets

Theorem
Let $V$ be a basic universe.

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

Let $\RR$ be a set.

Then $\Dom \RR$ and $\Img \RR$ are also sets.

Proof
From Domain of Relation is Subclass of Union of Union of Relation:
 * $\Dom \RR \subseteq \map \bigcup {\bigcup \RR}$

From Image of Relation is Subclass of Union of Union of Relation:
 * $\Img \RR \subseteq \map \bigcup {\bigcup \RR}$

We are given that $\RR$ is a set.

From the :
 * $\bigcup \RR$ is a set.

Applying the a second time:
 * $\map \bigcup {\bigcup \RR}$ is a set.

We have that $\RR \subseteq V$, by definition of basic universe

From the, it follows that every subclass of $V$ is a set.

The result follows.