Union Distributes over Intersection/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:
 * $\displaystyle \bigcap_{i \mathop \in I} \left({A_i \cup B}\right) = \left({\bigcap_{i \mathop \in I} A_i}\right) \cup B$

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

Also see

 * Intersection Distributes over Union/Family of Sets