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.

$\blacksquare$

Also see

• Mapping is Injection and Surjection iff Inverse is Mapping

Sources

• 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions