As a chart means it's a 1-1 diffeomorphism in the domain that defined the map. Let V=dom(\phi), U=f(V), W=\phi^{-1}(V)

Since f is injective in V, f is a 1-1 diffeomorphism between V and f(V).

And \phi is defined to be a 1-1 diffeomorphism between W and \phi(W)=V.

Compose two 1-1 diffeomorphism we get a 1-1 diffeomorphism.