Holonomic constraints, independent generalized coordinates

Consider a system of $3N$ particles with $p$ holonomic constraints: $g_v( \vec{r}_1 , \vec{r}_2 , ... , \vec{r}_N , t ) = 0 \, , \quad v = 1,2, ... , p \, .$ Various textbooks state that the current spatial configuration of a system of $3N$ particles with $p$ holonomic constraints is determined by $3N-p$ independent coordinates $q_1,...,q_{3N-p}$ . Independent means that there is no relation of the form $F(q_1,...,q_{3N-p}) = 0 \, .$ Unfortunately my textbooks do not provide a formal proof to this statement, so I am trying to figure out a proof by myself. These are my thoughts: At the time $t_0$ , the spatial configuration of our system is determined by a certain point of the set $M := \left\{ \vec{r} \in \mathbb{R}^{3N} \;\middle|\; \vec{f} ( \vec{r} ) = \vec{0} \right\} \, , \quad \vec{f} ( \vec{r}) := \begin{pmatrix} g_1 ( \vec{r} ,t_0) \\ g_2 ( \vec{r},t_0 ) \\ \vdots \\ g_p ( \vec{r} ,t_0) \end{pmatrix} \, .$ Now according to the preimage theorem, $M$ is a submanifold of $\mathbb{R}^{3N}$ of dimension $S := \text{dim} \, M = \text{dim} \, \mathbb{R}^{3N} - \text{dim} \, \mathbb{R}^{p} = 3N-p \, .$ Now according to definition, for every point $\vec{r}_0 \in M$ , there exists an open neighborhood $U$ and a diffeomorphism $\varphi : U \to V$ onto an open subset $V$ of a certain $\mathbb{R}^n$ , so that it holds: $\varphi ( M \cap U ) = \mathbb{R}_0^S \cap V \, , \quad \mathbb{R}_0^S := \left\{ \vec{q} \in \mathbb{R}^n \mid q_{S+1} = ... = q_n = 0 \right\} \, .$ So for every point $\vec{r} \in M \cap U$ , $\varphi$ is of the form $\varphi(\vec{r}) = ( q_1 , ... , q_S , 0 , ... , 0 ) \, .$ At this point I am not sure how to complete the proof, or whether this approach is even correct. Especially I do not know how to show that the $3N-p$ generalized coordinates are independent.