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

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.