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
From Inverse is Mapping implies Mapping is Injection and Surjection:


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

Again from Inverse is Mapping implies Mapping is Injection and Surjection:


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

The result follows by definition of bijection.

Also see

 * Mapping is Injection and Surjection iff Inverse is Mapping