Union of Inverses of Mappings is Inverse of Union of Mappings

Theorem

Let $I$ be an indexing set.

Let $\family {f_i: i \in I}$ 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 $\family {f_i: i \in I}$ is the union of the inverses of $f_i, i \in I$.

That is:

$\ds \paren {\bigcup \family {f_i: i \in I} }^{-1} = \bigcup \family {f_i^{-1}: i \in I}$

$\ds \tuple {y, x}$

$\in$

$\ds \paren {\bigcup \family {f_i: i \in I} }^{-1}$

$\ds \leadstoandfrom \ \ $

$\ds \tuple {x, y}$

$\in$

$\ds \family {f_i: i \in I}$

$\ds \leadstoandfrom \ \ $

$\ds \exists i \in I: \, $

$\ds \tuple {x, y}$

$\in$

$\ds f_i$

$\ds \leadstoandfrom \ \ $

$\ds \exists i \in I: \, $

$\ds \tuple {y, x}$

$\in$

$\ds f_i^{-1}$

$\ds \leadstoandfrom \ \ $

$\ds \tuple {y, x}$

$\in$

$\ds \bigcup \family {f_i^{-1}: i \in I}$

$\blacksquare$