# Preimage of Intersection under Mapping/Family of Sets

## Theorem

Let $S$ and $T$ be sets.

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

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

Then:

$\ds f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$

where:

$\ds \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$.
$f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.

## Proof 1

As $f$ is a mapping, it is by definition also a many-to-one relation.

It follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.

Thus Image of Intersection under One-to-Many Relation: Family of Sets can be applied for $\RR = f^{-1}$:

$\ds \RR \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} \RR \sqbrk {T_i}$

where $\RR \sqbrk {T_i}$ denotes the image of $T_i$ under $\RR$.

$\blacksquare$

## Proof 2

 $\ds x$ $\in$ $\ds f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i}$ $\ds \leadstoandfrom \ \$ $\ds \map f x$ $\in$ $\ds \bigcap_{i \mathop \in I} T_i$ $\ds \leadstoandfrom \ \$ $\ds \forall i \in I: \,$ $\ds \map f x$ $\in$ $\ds T_i$ $\ds \leadstoandfrom \ \$ $\ds \forall i \in I: \,$ $\ds x$ $\in$ $\ds f^{-1} \sqbrk {T_i}$ $\ds \leadstoandfrom \ \$ $\ds x$ $\in$ $\ds \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$

$\blacksquare$