Set Union is Self-Distributive/Families of Sets

Theorem
Let $I$ be an indexing set.

Let $\left \langle {A_i} \right \rangle_{i \mathop \in I}$ and $\left \langle {B_i} \right \rangle_{i \mathop \in I}$ be families of subsets of a set $S$.

Then:
 * $\displaystyle \bigcup_{i \mathop \in I} \left({A_i \cup B_i}\right) = \left({\bigcup_{i \mathop \in I} A_i}\right) \cup \left({\bigcup_{i \mathop \in I} B_i}\right)$

where $\displaystyle \bigcup_{i \mathop \in I} A_i$ denotes the union of $\left \langle {A_i} \right \rangle$.