\documentclass{book}
\usepackage[makeindex,nomain]{glossaries-extra}
\setglossarystyle{long3col}
\newcommand{\binding}[2]{#1\nolinebreak{\mapsto}\nolinebreak#2}
\newcommand{\finiteMap}[1]{\mathopen{\glsuseri{not:finiteMap}}#1\mathclose{\glsuserii{not:finiteMap}}}%%% Finite map, e.g., ⦃ x1↦a1, ... xn↦an ⦄.
\newglossary[nlg]{notation}{not}{ntn}{List of symbols}
\makeglossaries
\newglossaryentry{not:finiteMap}{type=notation, name={\(\{|\dotso|\}\)}, text={\{|\dotso|\}}, sort={finiteMap}, description={A finite map: \(\{| \binding{c_1}{s_1},\,\dotsc,\,\binding{c_n}{s_n}|\}\) is a function that maps \(c_i\) to \(s_i\) for each \(i\).},user1={\{|},user2={|\}}}
\begin{document}
\(\finiteMap{\binding{x}{a}}\)
\backmatter
\glsaddallunused
\printglossaries
\end{document}