Preimage of Intersection under Mapping/General Result

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

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

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.

Then:
 * $\ds f^{-1} \sqbrk {\bigcap \mathbb T} = \bigcap_{X \mathop \in \mathbb T} f^{-1} \sqbrk X$

Proof
$f$ is a mapping.

Therefore it is by definition 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 Image of Intersection under One-to-Many Relation: General Result applies:
 * $\ds \RR \sqbrk {\bigcap \mathbb T} = \bigcap_{X \mathop \in \mathbb T} \RR \sqbrk X$

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