How can I stack these items on top of each others at the bottom of a vertical bar?

Question

All my attempts have failed so far. I have the following:

\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
y &= y_0 + \sum_{n=1}^{\infty}  \frac{ x^n}{n!}  \, f_n(x,y) \bigg\vert_{x=x_0,y=y_0}
\end{align*}

I tried align* there, but LaTeX complained.

\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
y &= y_0 + \sum_{n=1}^{\infty}  \frac{ x^n}{n!}  \, f_n(x,y) \bigg\vert_{\begin{align*}x&=x_0\\y&=y_0\end{align*}}
\end{align*}

\end{document}

Package amsmath Error: Erroneous nesting of equation structures;(amsmath) trying to recover with `aligned'. \end{align*}

I am using LuaTeX in TeX Live 2022. I have reached the end of my LaTeX skill level with this.

Answer 1

I would just use \substack:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
  y &= y_0 + \sum_{n = 1}^{\infty} \frac{x^n}{n!} \, f_n(x, y) \bigg\vert_{\substack{x =x_0 \\ y=y_0}}
\end{align*}
    
\end{document}