Preimage of Intersection under Relation/Family of Sets

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

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

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

Then:
 * $\displaystyle \mathcal R^{-1} \left({\bigcap_{i \mathop \in I} S_i}\right) \subseteq \bigcap_{i \mathop \in I} \mathcal R^{-1} \left({S_i}\right)$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\left\langle{S_i}\right\rangle_{i \in I}$.

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