Preimage of Union 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 relation.

Then:
 * $\displaystyle f^{-1} \left[{\bigcup_{i \mathop \in I} T_i}\right] = \bigcup_{i \mathop \in I} f^{-1} \left[{T_i}\right]$

Proof
As $f$, being a mapping, is also a relation, we can apply Preimage of Union: Family of Sets:


 * $\displaystyle \mathcal R^{-1} \left[{\bigcup_{i \mathop \in I} T_i}\right] = \bigcup_{i \mathop \in I} \mathcal R^{-1} \left[{T_i}\right]$