Diagonal Relation is Functional

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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:

$\left({x, y_1}\right) = \left({x, x}\right)$
$\left({x, y_2}\right) = \left({x, x}\right)$

and so $y_1 = y_2$.

$\blacksquare$


Also see


Sources