Mapping from Finite Set to Itself is Injection iff Surjection
Let $S$ be a finite set.
Let $f: S \to S$ be a mapping.
Let $f$ be an injection.
We are given that $S$ is finite.
It follows from Infinite Set is Equivalent to Proper Subset that $\Img f = S$.
It follows by definition that $f$ is surjective.
Let $f$ be a surjection.
- $f \circ g = I_S$
where $I_S$ is the identity mapping.
By the above, it follows that $g$ is also a surjection.
Thus $g$ is a bijection.