Image of Intersection under Relation

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

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

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

Then:
 * $\mathcal R \left[{S_1 \cap S_2}\right] \subseteq \mathcal R \left[{S_1}\right] \cap \mathcal R \left[{S_2}\right]$

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

Also see

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

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.