There's a theorem named 'Global Rank theorem' in Lee's smooth manifold textbook which states that:
Let $M$ and $N$ be smooth manifolds, and suppose $F:M\to N$ is a smooth map of constant rank. (1) If $F$ is surjective then it's a smooth submersion. (2) If $F$ is injective then it's a smooth immersion. (3) If $F$ is bijective then it's a diffeomorphism.
The proof of (1) uses the Baire category theorem which is not quite intuitive to me.
I wonder if this argument works: Since $F$ has a constant rank, say $r$, if $p\in M$ then we can find a smooth chart $(U,\varphi)$ at $p$ and $(V,\psi)$ at $F(p)$ s.t. with respect to these coordinates, $F$ has a representation $\hat{F}(x_1,…,x_r,…,x_m)\to (x_1,…,x_r,0,…,0)$ where $m$ is the dimension of $M$ (and denote the dimension of $N$ by $n$). Since $\varphi,\psi$ are bijective, $\hat{F}$ should be surjective by assumption. Hence, $r = n$.
Hence, taking the differentials to $\varphi,\psi,\hat{F}$, we conclude $dF_p$ is surjective.
A similar argument shows (2). Does it work?
Proof given in Lee's textbook.
Best Answer
I think it does not work. The problem is that F surjective does not implies in general $ \hat{F}$ to being surjective. You are considering open sets when you are taking $\hat{F}$ and could happens that if you take $y$ in the codomain of $\hat{F}$, then there exists $x$ such that $F(x)=y$ but $x$ does not belong to the domain of $\hat{F}$.
I’m pretty sure that the main idea of the proof is that one that you explained, but you should fix this problem firstly.