2.5.4 The Characteristic Embedding of a Set
Let $X$ be a set.
The characteristic embedding1 of $X$ into $\mathcal{P}\webleft (X\webright )$ is the function
\[ \chi _{\webleft (-\webright )}\colon X \hookrightarrow \mathcal{P}\webleft (X\webright ) \]
defined by2
\begin{align*} \chi _{\webleft (-\webright )}\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{x}\\ & = \webleft\{ x\webright\} \end{align*}
for each $x\in X$.
Let $f\colon X\to Y$ be a map of sets.
-
Interaction With Functions. We have
Item 1: Interaction With Functions
Indeed, we have
\begin{align*} \webleft [f_{*}\circ \chi _{X}\webright ]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{*}\webleft (\chi _{X}\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{*}\webleft (\webleft\{ x\webright\} \webright )\\ & = \webleft\{ f\webleft (x\webright )\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{X'}\webleft (f\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\chi _{X'}\circ f\webright ]\webleft (x\webright ),\end{align*}
for each $x\in X$, showing the desired equality.
