# Image of Intersection under Relation/General Result

## Theorem

Let $S$ and $T$ be sets.

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

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

$\displaystyle \mathcal R \left[{\bigcap \mathbb S}\right] \subseteq \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$

## Proof

 $\displaystyle \forall X \in \mathbb S: \bigcap \mathbb S$ $\subseteq$ $\displaystyle X$ $\quad$ Intersection is Subset: General Result $\quad$ $\displaystyle \implies \ \$ $\displaystyle \forall X \in \mathbb S: \mathcal R \left[{\bigcap \mathbb S}\right]$ $\subseteq$ $\displaystyle \mathcal R \left[{X}\right]$ $\quad$ Image of Subset is Subset of Image $\quad$ $\displaystyle \implies \ \$ $\displaystyle \mathcal R \left[{\bigcap \mathbb S}\right]$ $\subseteq$ $\displaystyle \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$ $\quad$ Intersection is Largest Subset: General Result $\quad$

$\blacksquare$