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 $$S_i \subseteq S: i \in \mathbb{N}^*_n$$.

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