Mapping from Finite Set to Itself is Injection iff Surjection

Theorem
Let $S$ be a finite set.

Let $f: S \to S$ be a mapping.

Then $f$ is injective $f$ is surjective.

Proof
Let $f$ be an injection.

From Injection to Image is Bijection, $S$ is equivalent to the image $f \left({S}\right)$ of $f$.

We are given that $S$ is finite.

It follows from Infinite Set Equivalent to Proper Subset that $f \left({S}\right) = S$.

From Surjection iff Image equals Codomain it follows that $f$ is surjective.

Let $f$ be a surjection.

Then by Surjection iff Right Inverse there exists a mapping $g: S \to S$ such that:
 * $f \circ g = I_S$

where $I_S$ is the identity mapping.

By Right Inverse Mapping is Injection, $g$ is an injection.

By the above, it follows that $g$ is also a surjection.

Thus $g$ is a bijection.

It follows that $f = g^{-1}$ is also a bijection and so by definition an injection.