I'm looking for a text which contains a proof for the change-of-variables formula. Standard texts on calculus typically cover the case of a bijective $C^1$ change-of-variables, namely
\begin{equation}\label{eq:1}
\int_{F(U)} \phi(x)\, dx = \int_U( \phi \circ F )(y) \,|J_F|(y) \,dy.\tag{1}
\end{equation}
(here $J_F$ is the Jacobian of $F$).
I am looking for a reference for a the generalization for a non-injecitve, but still $C^1$, change-of-variables, namely
\begin{equation}\label{eq:2}
\int_{F(U)} \#(F^{-1}(x)) \,\phi(x)\,dx = \int_U( \phi \circ F )(y)\, |J_F|(y)\,dy.\tag{2}
\end{equation}
It appears that most text which offer a generalization of \eqref{eq:1} go far beyond \eqref{eq:2}: some include \eqref{eq:2} for Lipschitz maps, while the most common generalization of \eqref{eq:1} one finds is the (co-)area formula. Both are (significantly?) more technical than the simpler case of $C^1$ change-of-variables. Some books do mention formula \eqref{eq:2}, but as an exercise or remark after discussing \eqref{eq:1}.
My question is this: are there textbooks where formula \eqref{eq:2} is proven (at least in the form of a guided exercise) for the case of $C^1$ maps?
Note that this is not a exactly duplicate of this post, where the discussion is at least in part more pedagogical.
Thanks in advance!
