Image of Union under Relation/General Result

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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

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

$\blacksquare$