Image of Set Difference under Mapping/Corollary 1

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

Let $S_1 \subseteq S_2 \subseteq S$.

Then:
 * $\complement_{f \left[{S_2}\right]} \left({f \left[{S_1}\right]}\right) \subseteq f \left[{\complement_{S_2} \left({S_1}\right)}\right]$

where $\complement$ (in this context) denotes relative complement.

Proof
From Image of Set Difference under Relation: Corollary 1 we have:
 * $\complement_{\mathcal R \left[{S_2}\right]} \left({\mathcal R \left[{S_1}\right]}\right) \subseteq \mathcal R \left[{\complement_{S_2} \left({S_1}\right]}\right)$

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:
 * $\complement_{f \left[{S_2}\right]} \left({f \left[{S_1}\right]}\right) \subseteq f \left[{\complement_{S_2} \left({S_1}\right)}\right]$