Defining `\subsetcong` and `\subsetsim` via proper tilde over and under `\subset`

Question

I'd like to define a new command \subsetcong that would typeset a tilde over \subseteq to make it look similar to how tilde is over = in \cong. And also, I'd like to define \subsetsim to mean a \subset with a like tilde under, similar to \subsetapprox.

I am aware of this post: Big tilde in math mode, but the answer does not satisfy me, it feels needlessly complex. 'Detexify' app, didn't help much either, but I found one more symbol, which illustrates what I want with \subsetsim: \lesssim, only instead of < I need \subset.

The idea is to use A \subsetcong B or A \subsetsim B (yet to be decided), to mean something along the lines "there exists sub-element of B isomorphic (in a context-dependent sense) to A".

Any suggestions?

Best regards.

Answer 1

Something like this?

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\subsetcong}{\mathrel{\mathpalette\subset@cong\relax}}
\newcommand{\subsetsim}{\mathrel{\mathpalette\subset@sim\relax}}

\newcommand{\subset@cong}[2]{%
  \vbox{\offinterlineskip\m@th
    \ialign{\hfil$#1##$\hfil\cr
      \sim\cr\subset\cr
    }%
  }%
}
\newcommand{\subset@sim}[2]{%
  \vtop{\offinterlineskip\m@th
    \ialign{\hfil$#1##$\hfil\cr
      \subset\cr\noalign{\kern1pt}\sim\cr
    }%
  }%
}
\makeatother

\begin{document}

$A\subsetcong B\subsetsim C$

$\scriptstyle A\subsetcong B\subsetsim C$

$\scriptscriptstyle A\subsetcong B\subsetsim C$

\end{document}

Some refinements as per your comment, adjust to suit.

\documentclass{article}
\usepackage{amsmath,graphicx}

\makeatletter
\newcommand{\subsetcong}{\mathrel{\mathpalette\subset@cong\relax}}
\newcommand{\subsetsim}{\mathrel{\mathpalette\subset@sim\relax}}

\newcommand{\subset@cong}[2]{%
  \vbox{\offinterlineskip\m@th
    \ialign{\hfil##\cr
      \scalebox{0.9}{$#1\sim$}\cr
      \noalign{%
        \ifx#1\displaystyle\kern-0.5pt\else
        \ifx#1\textstyle\kern-0.5pt\fi\fi
      }%
      $#1\subset$\cr
    }%
  }%
}
\newcommand{\subset@sim}[2]{%
  \vtop{\offinterlineskip\m@th
    \ialign{\hfil##\cr
      $#1\subset$\cr\noalign{\kern0.5pt}\scalebox{0.9}{$#1\sim$}\cr
    }%
  }%
}
\makeatother

\begin{document}

$A\subsetcong B\subsetsim C$

$\scriptstyle A\subsetcong B\subsetsim C$

$\scriptscriptstyle A\subsetcong B\subsetsim C$

\end{document}