Preimage of Intersection under Relation

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

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

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

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