# 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:

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

## Proof

 $\ds \tuple {y, x}$ $\in$ $\ds \paren {\bigcup \family {f_i: i \in I} }^{-1}$ $\ds \leadstoandfrom \ \$ $\ds \tuple {x, y}$ $\in$ $\ds \family {f_i: i \in I}$ Definition of Inverse Relation $\ds \leadstoandfrom \ \$ $\ds \exists i \in I: \,$ $\ds \tuple {x, y}$ $\in$ $\ds f_i$ Definition of Union of Family $\ds \leadstoandfrom \ \$ $\ds \exists i \in I: \,$ $\ds \tuple {y, x}$ $\in$ $\ds f_i^{-1}$ Definition of Inverse Relation $\ds \leadstoandfrom \ \$ $\ds \tuple {y, x}$ $\in$ $\ds \bigcup \family {f_i^{-1}: i \in I}$ Definition of Union of Family

$\blacksquare$