Uniqueness Condition for Relation Value

Theorem
Let $\mathcal R$ be a relation.

Suppose that $\left({ x,y }\right) \in \mathcal R$.

Furthermore, suppose that $\exists ! y: \left({ x,y }\right) \in \mathcal R$.

Then, $\mathcal R \left({ x }\right) = y$, where $\mathcal R \left({ x }\right)$ denotes the value of $R$ at $x$.

If $y$ is not unique, then $\mathcal R \left({ x }\right) = \varnothing$.

Proof
Conversely:

Generalizing, $\forall z: \left({ z \in y \iff z \in \mathcal R \left({ x }\right) }\right)$.

Therefore, $y = \mathcal R \left({ x }\right)$ by the definition of class equality.

Suppose that $\neg \exists ! y: \left({ x,y }\right) \in \mathcal R$:

Thus, $\forall z: z \notin \mathcal R \left({ x }\right)$.

Therefore, $\mathcal R \left({ x }\right) = \varnothing$.