$S$ is a regular surface of the space and I need to prove that if $X:U\subset\mathbb{R}^2\rightarrow S$ is a parametrization of $S$, then $X:U\subset\mathbb{R}^2\rightarrow X(U)$ is a diffeomorphism.
The book gave a brief explanation, but I don't get it. It says that if I take another parametrization $X$ and use the fact that composition of $X^{-1}$ and $Y$ be a diffeomorphism, then $U$ and $X(U)$ are diffeomorphics.
Obs. The book definition of parametrization is that $X$ is homemorphism to his image with injective diferetial in all points of his domain.