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:
 * $$\left({t, s_1}\right) \in f^{-1} \and \left({t, s_2}\right) \in f^{-1} \implies s_1 = s_2$$

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