Image of Intersection under Relation

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

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

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

Proof
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Proof of General Result
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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 always holds iff $$\mathcal R$$ is one-to-many.