Let $A\subset {\mathbb R}^n $ be open. Let $\psi\colon {\mathbb R}^m \rightarrow {\mathbb R}^n$, $m<n$, a smooth, injective map of maximal rank $m$. Set $Y=\psi(A)$ and $Y_{\psi}$ the corresponding manifold. Now assume that $Y$ is given also implicitly by $\{x\in {\mathbb R}^{n}|f(x)=0\}$ where $f\colon {\mathbb R}^{n} \rightarrow {\mathbb R}^{n-m}$ is a smooth map.
Is it true that in this case $f$ has a maximal rank $n-m$ at each point $x\in Y$?
If possible please kindly restrict the argument to be suitable for a graduate student that took a course in multivariate calculus but not in differential geometry/topology.
Thanks.
Counter example: Take $f$ as above and $g=f^2$. Then $Y=\{x|g(x)=0\}$ but $Dg(y)$ is the zero matrix at all points $y\in Y$.