Inverse of Injection is Many-to-One Relation

Theorem
Let $f: S \to T$ be an injection.

Let $f^{-1}: T \to S$ be the inverse relation of $f$.

Then $f^{-1}$ is functional.

Proof
Let $f: S \to T$ be an injection.

We have by definition of inverse relation that:
 * $f^{-1} = \left\{{\left({t, s}\right): t = f \left({s}\right)}\right\}$

Let $f: S \to T$ be an injection.

Let $\left({t, s_1}\right) \in f^{-1}$ and $\left({t, s_2}\right) \in f^{-1}$.

By definition, we have that $f \left({s_1}\right) = t = f \left({s_2}\right)$.

But as $f$ is an injection:
 * $f \left({s_1}\right) = f \left({s_2}\right) \implies s_1 = s_2$

So we have that:
 * $\left({t, s_1}\right) \in f^{-1} \land \left({t, s_2}\right) \in f^{-1} \implies s_1 = s_2$

and so by definition, $f^{-1}$ is a functional relation.