# Intersection of Family is Subset of Intersection of Subset of Family

## Theorem

Let $I$ be an indexing set.

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

Let $J \subseteq I$.

Then:

$\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$

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

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Intersection is Subset $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in J: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Subset: $J \subseteq I$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in J} A_\alpha$ Definition of Intersection of Family

$\blacksquare$