# Injection to Image is Bijection

## Theorem

Let $f: S \rightarrowtail T$ be an injection.

Let $X \subseteq T$ be the image of $f$.

Then the restriction $f {\restriction_{S \times X}}: S \to X$ of $f$ to the image of $f$ is a bijection of $S$ onto $X$.

## Proof

We have:

Restriction of Injection is Injection
Restriction of Mapping to Image is Surjection

Thus we have that:

$f {\restriction_{S \times X}}: S \to X$ is an injection

and

$f {\restriction_{S \times X}}: S \to X$ is a surjection

Hence the result by definition of bijection.

$\blacksquare$