# Inverse Mapping is Bijection

## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ and $g: T \to S$ be inverse mappings of each other.

Then $f$ and $g$ are bijections.

## Proof

$f$ is both an injection and a surjection.
$g$ is both an injection and a surjection.

The result follows by definition of bijection.

$\blacksquare$