Union of Inverses of Mappings is Inverse of Union of Mappings

Theorem
Let $I$ be an indexing set.

Let $\left\{ {f_i: i \in I}\right\}$ be an indexed family of mappings.

For each $i \in I$, let $f^{-1}$ denote the inverse of $f$.

Then the inverse of the union of $\left\{ {f_i: i \in I}\right\}$ is the union of the inverses of $f_i, i \in I$.

That is:
 * $\displaystyle \left({\bigcup \left\{ {f_i: i \in I} \right\} }\right)^{-1} = \bigcup \left\{ {f_i^{-1}: i \in I} \right\}$