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.

iff:


 * the inverse $f^{-1}$ of $f$ is such that:
 * for each $y \in T$, the preimage $f^{-1} \left({\left\{{y}\right\}}\right)$ has exactly one element.


 * That is, such that $f^{-1} \subseteq T \times S$ is itself a mapping.