Existence and Uniqueness of Domain of Relation

Theorem
Let $V$ be a basic universe.

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

Then the domain $\Dom \RR$ of $\RR$ exists and is unique.

Proof
By the Axiom of Specification we can create the subclass of $V$:
 * $\Dom \RR = \set {x \in V: \exists y \in V: \tuple {x, y} \in \RR}$

Hence $\Dom \RR$ exists.

Suppose $\QQ \subseteq V$ such that $\QQ$ and $\Dom \RR$ are both the domain of $\RR$.

Then:
 * $\QQ = \set {x \in V: \exists y \in V: \tuple {x, y} \in \RR}$

Thus:
 * $x \in \QQ \implies x \in \Dom \RR$

and:
 * $x \in \Dom \RR \implies x \in \QQ$

Hence by the Axiom of Extension:
 * $\QQ = \Dom \RR$

and uniqueness has been demonstrated.