Preimage of Intersection 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:
 * $\displaystyle f^{-1} \left({\bigcap \mathbb T}\right) = \bigcap_{X \mathop \in \mathbb T} f^{-1} \left({X}\right)$

Proof
As $f$, being a mapping, is also a many-to-one relation, it follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.

Thus we can apply One-to-Many Image of Intersections/General Result:
 * $\displaystyle \mathcal R \left({\bigcap \mathbb T}\right) = \bigcap_{X \mathop \in \mathbb T} \mathcal R \left({X}\right)$

where here $\mathcal R = f^{-1}$.