Preimage of Union under Relation/Family of Sets

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

Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.

Let $\RR \subseteq S \times T$ be a relation.

Then:
 * $\displaystyle \RR^{-1} \sqbrk {\bigcup_{i \mathop \in I} T_i} = \bigcup_{i \mathop \in I} \RR^{-1} \sqbrk {T_i}$

where:
 * $\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\family {T_i}_{i \mathop \in I}$
 * $\RR^{-1} \left[{T_i}\right]$ denotes the preimage of $T_i$ under $\RR$.

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

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