Uniqueness Condition for Relation Value

Theorem
Let $\mathcal R$ be a relation.

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

Let:
 * $\exists ! y: \left({x, y}\right) \in \mathcal R$

Then:
 * $\mathcal R \left({x}\right) = y$

where $\mathcal R \left({x}\right)$ denotes the image of $\mathcal R$ 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 $\neg \exists ! y: \left({x, y}\right) \in \mathcal R$.

Then:

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

Therefore:
 * $\mathcal R \left({ x }\right) = \varnothing$