Set Intersection Preserves Subsets/Families of Sets

Theorem
Let $\left({A_i}\right)_{i \mathop \in I}, \left({B_i}\right)_{i \mathop \in I}$ be collections of sets.

Suppose that for all $i \in I: A_i \subseteq B_i$.

Then:


 * $\displaystyle \bigcap_{i \mathop \in I} A_i \subseteq \bigcap_{i \mathop \in I} B_i$

Proof
Suppose that $x \in \displaystyle \bigcap_{i \mathop \in I} A_i$.

That is, $x \in A_i$ for all $i \in I$, by definition of intersection.

Since $A_i \subseteq B_i$ for all $i \in I$ as well, it follows that $x \in B_i$ for all $i \in I$ by definition of subset.

Thus, again by definition of intersection, it follows that:


 * $\displaystyle x \in \bigcap_{i \mathop \in I} B_i$

Hence the result, by definition of subset.