# Image of Intersection under Relation/Family of Sets

## Theorem

Let $S$ and $T$ be sets.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

Let $\mathcal R \subseteq S \times T$ be a relation.

Then:

$\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right] \subseteq \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\left\langle{S_i}\right\rangle_{i \in I}$.

## Proof

 $\displaystyle \forall i \in I: \bigcap_{j \mathop \in I} S_j$ $\subseteq$ $\displaystyle S_i$ Intersection is Subset: Family of Sets $\displaystyle \implies \ \$ $\displaystyle \forall i \in I: \mathcal R \left[{\bigcap_{j \mathop \in I} S_j}\right]$ $\subseteq$ $\displaystyle \mathcal R \left[{S_i}\right]$ Image of Subset is Subset of Image $\displaystyle \implies \ \$ $\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right]$ $\subseteq$ $\displaystyle \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]$ Intersection is Largest Subset: Family of Sets

$\blacksquare$