Image of Union under Relation

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

The image of the union of subsets of $S$ is equal to the union of their images.

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

Then $\mathcal R \left({S_1 \cup S_2}\right) = \mathcal R \left({S_1}\right) \cup \mathcal R \left({S_2}\right)$.

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:
 * $\displaystyle \mathcal R \left({\bigcup \mathbb S}\right) = \bigcup_{X \in \, \mathbb S} \mathcal R \left({X}\right)$