Intersection Distributes over Union/Family of Sets

Theorem
Let $I$ be an indexing set.

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

Let $B \subseteq S$.

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

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