Is Rudin's proof of the inverse function theorem correct?

This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.).It defines for each $y \in \mathbb{R}^n$

$$\phi(x)=x+A^{-1}(y-f(x))$$

where $A$ is the matrix of the differential of $f$ at $a$. it shows that$$|\phi(x) - \phi(z)| \leq \dfrac{1}{2}|x-z|$$on some open ball centered on a such that $|f'(x)-A|< \lambda = (2|A^{-1}|)^{-1}$. Hence it states that it has at most one fixed point in $U$, so that $f(x) = y$ for at most one $x$ and $U$, so $f$ will be bijective in $U.$ For me this is not clear, as you used the contraction principle here, since $U$ is open, and it is not clear with what you have so far that $\phi(U) \subset U$?