Image of Subset under Relation is Subset of Image/Corollary 1

Corollary to Image of Subset is Subset of Image
Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation from $S$ to $T$.

Let $C, D \subseteq T$.

Then:
 * $C \subseteq D \implies f_{\mathcal R^{-1}} \left({C}\right) \subseteq f_{\mathcal R^{-1}} \left({D}\right)$

where$\mathcal R^{-1}$ is the inverse of $\mathcal R$.

Proof
We have that $\mathcal R^{-1}$ is itself a relation, by definition of inverse relation.

The result follows directly from Image of Subset is Subset of Image.