Union of Inverses of Mappings is Inverse of Union of Mappings

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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:

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


\(\ds \tuple {y, x}\) \(\in\) \(\ds \paren {\bigcup \family {f_i: i \in I} }^{-1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \family {f_i: i \in I}\) 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