Restriction of Mapping to Image is Surjection
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 $f$.
- $\Img g \subseteq T$
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.