Set Intersection is Self-Distributive/Families of Sets

Theorem
Let $I$ be an indexing set.

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

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

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

Proof
Thus by definition of subset:
 * $\displaystyle \bigcap_{\alpha \mathop \in I} \left({A_\alpha \cap B_\alpha}\right) \subseteq \left({\bigcap_{\alpha \mathop \in I} A_\alpha}\right) \cap \left({\bigcap_{\alpha \mathop \in I} B_\alpha}\right)$

Thus by definition of subset:
 * $\displaystyle \left({\bigcap_{\alpha \mathop \in I} A_\alpha}\right) \cap \left({\bigcap_{\alpha \mathop \in I} B_\alpha}\right) \subseteq \bigcap_{\alpha \mathop \in I} \left({A_\alpha \cap B_\alpha}\right)$

By definition of set equality:


 * $\displaystyle \bigcap_{\alpha \mathop \in I} \left({A_\alpha \cap B_\alpha}\right) = \left({\bigcap_{\alpha \mathop \in I} A_\alpha}\right) \cap \left({\bigcap_{\alpha \mathop \in I} B_\alpha}\right)$