Uniqueness Condition for Relation Value

Theorem
Let $\RR$ be a relation.

Let $\tuple {x, y} \in \RR$.

Let:
 * $\exists ! y: \tuple {x, y} \in \RR$

Then:
 * $\map \RR x = y$

where $\map \RR x$ denotes the image of $\RR$ at $x$.

If $y$ is not unique, then:
 * $\map \RR x = \O$

Proof
Conversely:

Generalizing:
 * $\forall z: \paren {z \in y \iff z \in \map \RR x}$

Therefore:
 * $y = \map \RR x$

by the definition of class equality.

Suppose that $\neg \exists ! y: \tuple {x, y} \in \RR$.

Then:

Thus:
 * $\forall z: z \notin \map \RR x$

Therefore:
 * $\map \RR x = \O$