Union of Functions Theorem/Corollary

Theorem
Let $X$ be a set.

Let $\left\{{X_i: i \in \N}\right\}$ be an exhausting sequence of sets on $X$.

For each $i \in \N$, let $g_i: X_i \to Y$ be a mapping such that:
 * $g_{i+1} \restriction X_i = g_i$

where $g_{i+1} \restriction X_i$ denotes the restriction of $g_{i+1}$ to $g_i$. For each $i \in \N$, let $g_i : X_i \to Y$ be invertible.

Then $\displaystyle \bigcup \left\{{g_i: i \in \N}\right\}$ is invertible and:
 * $\displaystyle \left({\bigcup \left\{{g_i: i \in \N}\right\} }\right)^{-1} = \bigcup \left\{{g_i^{-1}: i \in \N}\right\}$