Preimage of Intersection under Mapping/Family of Sets/Proof 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 $\mathcal R = f^{-1}$:
 * $\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} T_i}\right] = \bigcap_{i \mathop \in I} \mathcal R \left[{T_i}\right]$

where $\mathcal R \left[{T_i}\right]$ denotes the image of $T_i$ under $\mathcal R$.