Preimage of Union under Mapping/General Result

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

Let $f: S \to T$ be a mapping.

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

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

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

Proof
As $f$, being a mapping, is also a relation, we can apply Preimage of Union under Relation: General Result:


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