# Injection to Image is Bijection

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## 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:

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.

$\blacksquare$

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 8$: Example $8.2$