Restriction of Mapping to Image is Surjection

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

Let $g: S \to \operatorname{Im} \left({f}\right)$ be the restriction of $f$ to $S \times \operatorname{Im} \left({f}\right)$.

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

Proof
The fact that $f: \operatorname{Dom} \left({f}\right) \to \operatorname{Im} \left({f}\right)$ is a surjection follows directly from Surjection iff Image equals Codomain.

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.