If $f:M\rightarrow N$ is $C^{\infty}$, bijective, and everywhere non-singular, then $f$ is a diffeomorphism

Question

I am not able to solve this problem:

Prove that if $f:M\rightarrow N$ is $C^{\infty}$, one-to-one, onto, and everywhere non-singular, then $f$ is a diffeomorphism.

This $f$ is a diffeomorphism $\iff$ $df$ is surjective everywhere, right? Then how to proceed?