# 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:

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

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

## Proof

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

$\blacksquare$