# Image of Intersection under Mapping/Family of Sets

## Theorem

Let $S$ and $T$ be sets.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Let $f: S \to T$ be a mapping.

Then:

$\displaystyle f \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} f \sqbrk {S_i}$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\family {S_i}_{i \mathop \in I}$.

## Proof

As $f$, being a mapping, is also a relation, we can apply Image of Intersection under Relation: Family of Sets:

$\displaystyle \mathcal R \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} \mathcal R \sqbrk {S_i}$

$\blacksquare$