If $M$ and $N$ are manifolds and $F:M→N$ is bijective, smooth and restricts to a diffeomorphism on submanifolds of $M$, is $F$ a diffeomorphism?

Question

Let $M$ and $N$ be smooth manifolds and let $F:M\rightarrow N$ be a bijective smooth map. Suppose that for every point $p\in M$, there exists a regular submanifold $M_{p}$ of M containing $p$ and a regular submanifold $N_{F\left(p\right)}$ of $N$ containing $F(p)$ such that $F$ maps $M_{p}$ bijectively to $N_{F\left(p\right)}$ and the map $F_{p}:M_{p}\rightarrow N_{F\left(p\right)}$ defined by $F_{p}\left(x\right)=F\left(x\right)$ for all $x\in M_{p}$ is a diffeomorphism, does it follows that $F$ is a diffeomorphism?

I am trying to use this result to show that a bijective morphism of smooth vector bundles is invertible.

Answer 1

Consider $\mathbb{R}^2$ and the map defined by $f(x,y)=(x^3,y)$. $f$ is bijective, For every $(x_0,y_0)$, take the line $(x_0,y)$ the restriction of $f$ on this line is a diffeo onto its image, but $f^{-1}$ is not differentiable at $(0,0)$.