Inverse is Mapping implies Mapping is Injection and Surjection

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

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

Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.

Then:
 * $(1): \quad f$ is an injection
 * $(2): \quad f$ is a surjection.

Proof
This is divided into two parts: