Preimage of Union under Relation/General Result

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:
 * $\displaystyle \RR^{-1} \sqbrk {\bigcup \mathbb T} = \bigcup_{X \mathop \in \mathbb T} \RR^{-1} \sqbrk X$

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.