If bundle map is a local isomorphism, then bundle map is diffeomorphic?

Question

Suppose $\phi$ is a bundle map from $E$ to $F$, where $E$ and $F$ are two vector bundles over a manifold $M$. Now for each $x\in M$, $\phi$ restricted to $E_x:E_x\to F_x$ is a isomorphism, how to show that $\phi$ indeed a diffeomorphism?

Answer 1

Using the fact that $\phi$ is bijective on the fibers, show that it is in fact bijective. To show it is a diffeo then, it is enough to do it locally on $M$. You can therefore assume that both $E$ and $F$ are trivial. Can you see how to proceed?