Implicit Function Theorem says if $F_i(x_1,..,x_n,y_1,..,y_n)=0,(i=1..n)$, and $\det\left(\frac{\partial F_i}{\partial x_j}\right)\neq0$, then $x_i$ can be expressed in terms of $\{y_j\}$. If $\det\left(\frac{\partial F_i}{\partial x_j}\right)=0$, $x_i$ can't be expressed in terms of $\{y_j\}$.
My question is, if $y_i$ can be expressed in terms of $\{x_i\}$, and $\det\left(\frac{\partial F_i}{\partial x_j}\right)=0$, then there exists $(n-r)$ functions $\phi_m(y_1,..,y_n)=0,(m=1,..,n-r)$, how to prove it? I read its proof in a physics book, but it's not nicely proved.
ps.$r$ stands for rank of $\frac{\partial F_i}{\partial x_j}$.
