Mapping is Injection and Surjection iff Inverse is Mapping

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Then:
 * $f: S \to T$ can be defined as a bijection in the sense that:
 * $(1): \quad f$ is an injection
 * $(2): \quad f$ is a surjection




 * the inverse $f^{-1}$ of $f$ is itself a mapping.