I have not been able to track down a proof of the following theorem:
Let $h:(M,\mathcal{U},\mathcal{F}) \to (M',\mathcal{U}',\mathcal{F}')$ be a smooth mapping between differential manifolds (U - open cover, F - atlas of charts) of dimensions m and m'. Suppose that for some $q\in M'$, $dh_p:T_p M \to T_{h(p)} M'$ is surjective for any $p\in N = h^{-1}q$. Then $N\subseteq M$ is a submanifold of dimension m-m' and the tangent space at $p\in N$ is given by $T_p N = \text{ker}~dh_p$.
Can you help?