# Mapping is Injection and Surjection iff Inverse is Mapping

## Theorem

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.

## Proof

### Necessary Condition

Let $f: S \to T$ be a mapping such that:

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

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

### Sufficient Condition

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.