How to typeset a double subscript (sic!) not a subsubscript correctly

Question

I have a function fthat I want to restrict to a certain sub-domain of the domain it has been originally defined for. Usually, in mathematics this is illustrated by a vertical bar followed by the restricted set in the lower right of the function symbol. E.g. usually I write

f_{\mid A}

which means f restricted to the set A.

Unfortunately, I have a sequence of functions f_i that has already a subscript. If I write

f_{i_{{\mid A}}}

I do net get an LaTeX error, but the result is semantically wrong. \bar A is not a subscript of i which happens to be a subscript of f, but \mid A is a subscript of f_i. Especially, the font size of {\mid A} should not be smaller than the size of i, but have the same size.

However, if a write

{f_i}_{\mid A}

(which btw would be semantically correct) I get a "double subscript error".

Alternatively, I could write

f_{i \; \mid A}

and get the correct font size for \mid A but it is placed too high. \mid A should be below i.

MWE

\documentclass{article}
\usepackage[TS1,T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}

\begin{document}

\begin{equation}
{\hat{f}_i}_{\mid A}
\end{equation}

\end{document}

As I have already found out, the problem occurs only with the accent above f. But I need the accent there.

Answer 1

It is a known problem, see Why am I getting a double subscript error?:

{\hat{f}_i}_{|A}

will issue a “double subscript” error. The cure is adding {} to the base.

On the other hand, I can't see so much a difference between this and

\hat{f}_{i\,|A}

which I'd prefer.

Here's the code.

\documentclass{article}

\begin{document}

\begin{equation}
\hat{f}_{i\,|A}
\qquad
{{}\hat{f}_i}_{|A}
\end{equation}

\end{document}

Note that \mid A is semantically wrong, although leading to the same result.