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:
 * $\mathcal R \sqbrk {S_1 \cap S_2} \subseteq \mathcal R \sqbrk {S_1} \cap \mathcal R \sqbrk {S_2}$

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 under Relation
 * Preimage of Union under Relation
 * Preimage of Intersection under Relation

Also see

 * Image of Intersection under One-to-Many Relation, which shows that, for the general relation $\mathcal R$, equality holds $\mathcal R$ is one-to-many.