Preimage of Intersection under Relation

Theorem
Let $S$ and $T$ be sets.

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

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

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

Also see

 * Image of Intersection
 * Image of Union
 * Preimage of Union

Proof
This follows from Image of Intersection, and the fact that $\mathcal R^{-1}$ is itself a relation, and therefore obeys the same rules.