Intersection of Family is Subset of Intersection of Subset of Family

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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

$\blacksquare$


Also see


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