# Restriction of Mapping to Image is Surjection

## Theorem

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

Let $g: S \to \Img f$ be the restriction of $f$ to $S \times \Img f$.

Then $g$ is a surjective restriction of $f$.

## Proof

$\Img g \subseteq T$

Furthermore, by definition of image, we have:

$\forall s \in S: g \paren s \in \Img g$

Therefore, $g$ may be viewed as a mapping $g: S \to \Img g$.

Thus, by definition, $g$ is a surjection.

$\blacksquare$

## Comment

Thus, for any mapping $f: S \to T$ which is not surjective, by restricting its codomain to its image, it can be considered as a surjection.