Preimage of Union under Relation/General Result

Theorem
Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:
 * $\displaystyle \mathcal R^{-1} \left[{\bigcup \mathbb T}\right] = \bigcup_{X \mathop \in \mathbb T} \mathcal R^{-1} \left[{X}\right]$

where $\mathcal R^{-1} \left[{X}\right]$ denotes the preimage of $X$ under $\mathcal R$.

Proof
We have that $\mathcal R^{-1}$ is a relation.

The result follows from Image of Union under Relation: General Result.