Preimage of Union under Mapping/Family of Sets/Proof 2

Proof
We have that $f$ is a mapping, and so also a relation.

Thus its inverse $f^{-1}$ is also a relation.

Hence we can apply Image of Union under Relation: Family of Sets:


 * $\displaystyle \mathcal R \left[{\bigcup_{i \mathop \in I} T_i}\right] = \bigcup_{i \mathop \in I} \mathcal R \left[{T_i}\right]$

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