Preimage of Union under Relation/General Result
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Theorem
Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
- $\ds \RR^{-1} \sqbrk {\bigcup \mathbb T} = \bigcup_{X \mathop \in \mathbb T} \RR^{-1} \sqbrk X$
where $\RR^{-1} \sqbrk X$ denotes the preimage of $X$ under $\RR$.
Proof
We have that $\RR^{-1}$ is a relation.
The result follows from Image of Union under Relation: General Result.
$\blacksquare$