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 Axiom of Unions:

$\bigcup \RR$ is a set.

Applying the Axiom of Unions a second time:

$\map \bigcup {\bigcup \RR}$ is a set.

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

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

The result follows.

$\blacksquare$

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: Theorem $8.2$