I'd like to verify the following statement, which intuitively seems quite reasonable, by a rigorous proof:
Let $M \subset \mathbb{R}^D$ be a $d$-dimensional $C^1$ submanifold embedded in $\mathbb{R}^D$ and let $a \in M$. Further assume that the tangent plane of $M$ at $a$ is $T = span(e_1,...,e_d) \subset \mathbb{R}^D$, where $e_i$ denotes the i-th standard unit vector.
Then there exist a $C^1$ function $f: U \to V$ and open sets $U > \subset \mathbb{R}^d$, $V \subset \mathbb{R}^{D-d}$ $C^1$ function $f$, such that they contain the respective projection of a and $$ M \cap (U \times V) = \{ (x,f(x)): x \in U\} $$
It is standard in most textbooks to show that a submanifold can always be locally represented as a graph. However, in all versions of this theorem I could find it is not specified which of the $D$ coordinates will be used as an instrinsic coordinate and which ones will be just a function of those (it's always noted as $(x,f(x))$ but "up to a possible change of coordinates").
Thus, the main point here is, that if the tangent plane is the plane spanned by the first $d$ unit-vectors, then this plane can indeed be used as the domain for $f$.
I had the feeling that this should not be too hard to prove. However, I was not able to do it myself so far. I have looked into many proofs of the standard theorem that start with the representation of the manifold as the zero set of $F: \mathbb{R}^D \to \mathbb{R}^{D-d}$ and then show the existence of $f$. The crucial point there would be to show that the $D-d$ last column vectors of $F'(a)$ are linearly independant. However, I was not able to do this.
I'd be thankful for advice!
HINT: Can you show that the linear projection $\pi\colon M \to T$ is a local diffeomorphism at $a$? That is, can you apply the Inverse Function Theorem?