Image of Intersection under Relation

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

Theorem
The image of the intersection is a subset of the intersection of the images.

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

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

Generalized Result
Let $$S_i \subseteq S: i \in \N^*_n$$.

Then $$\mathcal{R} \left({\bigcap_{i = 1}^n S_i}\right) \subseteq \bigcap_{i = 1}^n \mathcal{R} \left({S_i}\right)$$.

Proof
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Generalized Proof
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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.