Preimage of Union under Relation/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 $\mathcal R \subseteq S \times T$ be a relation.

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

where $\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\left\langle{T_i}\right\rangle_{i \in I}$.

Proof
This follows from Image of Union/Family of Sets, and the fact that $\mathcal R^{-1}$ is itself a relation, and therefore obeys the same rules.