Image under Inclusion Mapping

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Theorem

Let $X$ be a set.

Let $S \subseteq X$, $Z \subseteq S$.


Then $i_S\left[{Z}\right] = Z$

where

$i_S$ denotes the inclusion mapping of $S$,
$i_S\left[{Z}\right]$ denotes the image of $Z$ under $i_S$.


Proof

Thus

\(\displaystyle i_S\left[{Z}\right]\) \(=\) \(\displaystyle \left\{ {i_S\left({z}\right): z \in Z}\right\}\) definition of image of set
\(\displaystyle \) \(=\) \(\displaystyle \left\{ {z: z \in Z}\right\}\) definition of inclusion mapping
\(\displaystyle \) \(=\) \(\displaystyle Z\) definition of set equality

$\blacksquare$

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