Image of Set Difference under Mapping/Corollary 2

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

Let $S_1$ and $S_2$ be subsets of $S$.

Then:
 * $\complement_{\operatorname{Im} \left({f}\right)} \left({f \left[{S_1}\right]}\right) \subseteq f \left[{\complement_S \left({S_1}\right)}\right]$

where $\operatorname{Im} \left({f}\right)$ denotes the image of $f$.

Proof
From Image of Set Difference under Relation: Corollary 2 we have:
 * $\complement_{\operatorname{Im} \left({\mathcal R}\right)} \left({\mathcal R \left[{S_1}\right]}\right) \subseteq \mathcal R \left[{\complement_S \left({S_1}\right)}\right]$

where $\mathcal R \subseteq S \times T$ is a relation on $S \times T$.

As $f$, being a mapping, is also a relation, it follows directly that:
 * $\complement_{\operatorname{Im} \left({f}\right)} \left({f \left[{S_1}\right]}\right) \subseteq f \left[{\complement_S \left({S_1}\right)}\right]$