Intersection Distributes over Union/Family of Sets

Theorem
Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be a indexed family of subsets of a set $S$.

Let $B \subseteq S$.

Then:
 * $\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.

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

By definition of subset:
 * $\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B \subseteq \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B}$

By definition of set equality:
 * $\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

Also see

 * Union Distributes over Intersection/Family of Sets