Image under Inclusion Mapping

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

$\blacksquare$