Diagonal Relation is Many-to-One

Theorem
The diagonal relation is many-to-one.

That is:
 * $\forall x \in \Dom {\Delta_S}: \tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S \implies y_1 = y_2$

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 $\tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S$.

From the definition of the diagonal relation:
 * $\tuple {x, y_1} = \tuple {x, x}$
 * $\tuple {x, y_2} = \tuple {x, x}$

and so $y_1 = y_2$.

Also see

 * Definition:Diagonal Mapping