Diagonal Relation is Many-to-One

Theorem
The diagonal relation is functional.

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

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

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