Preimage of Union under Relation/Family of Sets

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

Let $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ 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 $\left\langle{T_i}\right\rangle_{i \mathop \in I}$
 * $\mathcal R^{-1} \left[{T_i}\right]$ denotes the preimage of $T_i$ under $\mathcal R$.

Proof
We have that $\mathcal R^{-1}$ is itself a relation.

The result follows from Image of Union under Relation: Family of Sets.