I was looking at the following post: Prove the theorem of level sets as manifolds. And I am confused about how the implicit function theorem was used here. I understand that $\left( \frac{\partial f_i}{\partial x_j}(x) \right)$ had rank $r$, this follows from assumption as the rows are linearly independent. This means we can actually remove columns to make our matrix $r\times r$ and invertible. We keep $1\leq \alpha_1<\alpha_2<\dots<\alpha_r\leq n$ indexed columns and discard the $1\leq \alpha'_1<\alpha'_2<\dots<\alpha'_{n-r}\leq n$ indexed columns. Now for the implicit function theorem we need some function $g$ with a point $(x,y)$ s.t $g(x,y)=0$ and we require $Dg$ w.r.t $y$ variables to be invertible. What is the $g$ in the post above? What is $(x,y)?$
My first thought is that we just have to order our functions properly.
$g(x,y)=(f_{\alpha'_1},\dots,f_{\alpha'_{n-r}},f_{\alpha_1},\cdots,f_{\alpha_r})$ Then we have $g(a)=0$ but this seems wrong as now we have $g(a_1,\dots,a_n)=0$ and $Dg$ w.r.t. the last $r$ variables is not actually the matrix we want. Could someone give me how to apply the implicit function theorem here in detail?