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
Thus $f$ is by definition an injection.

Thus $f$ is by definition a surjection.

Similarly:

Thus $g$ is by definition an injection.

Thus $g$ is by definition a surjection.

So $f$ and $g$ are both injections and surjections.

The result follows by definition of bijection.

Also see

 * Mapping is Injection and Surjection iff Inverse is Mapping