Preimage of Set Difference under Relation

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

Let $C$ and $D$ be subsets of $T$.

Then:
 * $\mathcal R^{-1} \left[{C}\right] \setminus \mathcal R^{-1} \left[{D}\right] \subseteq \mathcal R^{-1} \left[{C \setminus D}\right]$

where:
 * $\setminus$ denotes set difference
 * $\mathcal R^{-1} \left[{C}\right]$ denotes the preimage of $C$ under $\mathcal R$.

Proof
We have that $\mathcal R^{-1}$ is itself a relation

The result then follows from Image of Set Difference under Relation.