Preimage of Intersection under Mapping/Family of Sets

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

Let $\left\langle{T_i}\right\rangle_{i \in I}$ be a family of subsets of $T$.

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

Then:
 * $\displaystyle f^\gets \left({\bigcap_{i \mathop \in I} T_i}\right) = \bigcap_{i \mathop \in I} f^\gets \left({T_i}\right)$

where:
 * $\displaystyle \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\left\langle{T_i}\right\rangle_{i \mathop \in I}$.
 * $f^\gets$ denotes the mapping induced on the power set $\mathcal P \left({T}\right)$ of $T$ by the inverse $f^{-1}$.

Proof
As $f$ is a mapping, it is by definition 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 Image of Intersection under One-to-Many Relation: Family of Sets can be applied for $\mathcal R = f^{-1}$:
 * $\displaystyle \mathcal R^\to \left({\bigcap_{i \mathop \in I} T_i}\right) = \bigcap_{i \mathop \in I} \mathcal R^\to \left({T_i}\right)$

where $\mathcal R^\to$ denotes the mapping induced on the power set $\mathcal P \left({T}\right)$ of $T$ by $\mathcal R$.