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.