Let $f:S \rightarrow \mathbb{R}$ where S is a regular surface be a differentiable function. Show that the inverse image of a regular value of $f $ is a regular curve on S.
My attempt : Let $a$ be a regular value of $f$and $p\in f^{-1}(a)$. Since $a$ is a regular value we have that $\nabla f \neq 0$ .Suppose wlog that $f_z\neq0$ at p .Define $F:S\rightarrow \mathbb{R^3} $ by $F(x,y,z) = (x,y,f(x,y,z))$ . Then the differential $dF_p$ is an invertible matrix .
Then by the Inverse Function Theorem we have that exist neighborhoods $V$ of $p$ and $W$ of $F(p)$ such that $F:V \rightarrow W$ is differentiable and $ F^{-1}:W\rightarrow V $ is invertible. From this point I tried to adapt the proof that the inverse image of regular value is a regular surface but it didn’t work .
Any hints on how to continue ?