Image of Union under Relation

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

The image of the union is equal to the union of the images.

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

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

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

Proof
$$ $$ $$ $$ $$

Generalized Proof
$$ $$ $$ $$ $$