Mapping to Image is Surjection

Corollary to Surjection iff Image equals Codomain
Every mapping $f: S \to T$ can be made into a surjection $f: S \to \operatorname{Im} \left({f}\right)$. Here, $\operatorname{Im} \left({f}\right)$ denotes the image of $f$.

Proof
From Image is Subset of Codomain, $\operatorname{Im} \left({f}\right) \subseteq T$.

Furthermore, by definition of image, we have:


 * $\forall s \in S: f \left({s}\right) \in \operatorname{Im} \left({f}\right)$

Therefore, $f$ may be viewed as a mapping $f: S \to \operatorname{Im} \left({f}\right)$.

That it is a surjection follows from the definition of image, and Surjection iff Image equals Codomain.