Preimage of Union under Mapping/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 $f: S \to T$ be a relation.

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

where:
 * $\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\family {T_i}_{i \mathop \in I}$
 * $f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.