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