# Image of Union 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:

$\displaystyle \mathcal R \left[{\bigcup \mathbb S}\right] = \bigcup_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$

## Proof

 $\displaystyle t$ $\in$ $\displaystyle \mathcal R \left[{\bigcup \mathbb S}\right]$ $\displaystyle \iff \ \$ $\displaystyle \exists s \in \bigcup \mathbb S: t$ $\in$ $\displaystyle \mathcal R \left[{s}\right]$ Image of Subset under Relation equals Union of Images of Elements $\displaystyle \iff \ \$ $\displaystyle \exists X \in \mathbb S: \exists s \in X: t$ $\in$ $\displaystyle \mathcal R \left[{s}\right]$ Definition of Union of Set of Sets $\displaystyle \iff \ \$ $\displaystyle \exists X \in \mathbb S: t$ $\in$ $\displaystyle \mathcal R \left[{X}\right]$ Definition of Image of Subset under Relation $\displaystyle \iff \ \$ $\displaystyle t$ $\in$ $\displaystyle \bigcup_{X \mathop \in \mathbb S} \mathcal R \left[{X}\right]$ Definition of Union of Set of Sets

$\blacksquare$