Image of Intersection under Relation

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

The image of the intersection of subsets of $S$ is a subset of the intersection of their images.

Let $S_1$ and $S_2$ be subsets of $S$.

Then $\mathcal R \left({S_1 \cap S_2}\right) \subseteq \mathcal R \left({S_1}\right) \cap \mathcal R \left({S_2}\right)$.

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

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:
 * $\displaystyle \mathcal R \left({\bigcap \mathbb S}\right) \subseteq \bigcap_{X \in \mathbb S} \mathcal R \left({X}\right)$

Note
Note that equality does not hold in general.

See the note on Mapping Image of Intersection for an example of a mapping (which is of course a relation) for which it does not.

Also see One-to-Many Image of Intersections, which shows that, for the general relation $\mathcal R$, equality holds iff $\mathcal R$ is one-to-many.