Intersection Distributes over Union/Family of Sets

Theorem
Let $I$ be an indexing set.

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

Let $B \subseteq S$.

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

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\left \langle {A_\alpha} \right \rangle_{\alpha \mathop \in I}$.

Proof
By definition of subset:
 * $\displaystyle \bigcup_{\alpha \mathop \in I} \left({A_\alpha \cap B}\right) \subseteq \left({\bigcup_{\alpha \mathop \in I} A_\alpha}\right) \cap B$

By definition of subset:
 * $\displaystyle \left({\bigcup_{\alpha \mathop \in I} A_\alpha}\right) \cap B \subseteq \bigcup_{\alpha \mathop \in I} \left({A_\alpha \cap B}\right)$

By definition of set equality:
 * $\displaystyle \bigcup_{\alpha \mathop \in I} \left({A_\alpha \cap B}\right) = \left({\bigcup_{\alpha \mathop \in I} A_\alpha}\right) \cap B$

Also see

 * Union Distributes over Intersection/Family of Sets