Diagonal Relation is Many-to-One

Theorem
The diagonal relation is functional.

That is:
 * $\forall x \in \operatorname{Dom} \left({\Delta_S}\right): \left({x, y_1}\right) \in \Delta_S \land \left({x, y_2}\right) \in \Delta_S \implies y_1 = y_2$

where $\Delta_S$ is the diagonal relation on a set $S$.

Proof
Let $S$ be a set and let $\Delta_S$ be the diagonal relation on $S$.

Let $\left({x, y_1}\right) \in \Delta_S \land \left({x, y_2}\right) \in \Delta_S$.

From the definition of the diagonal relation: and so $y_1 = y_2$.
 * $\left({x, y_1}\right) = \left({x, x}\right)$
 * $\left({x, y_2}\right) = \left({x, x}\right)$