I'm studying Do Carmo's differential geometry of curves and surfaces book and I have come up with the following question:
Let $S_1 \subset \mathbb{R}^3$ a regular surface and $S_2 \subset \mathbb{R}^3$ such that there exists a diffeomorphism $f:S_1 \rightarrow S_2$ (a differentiable and bijective function such that $f^{-1} :S_2 \rightarrow S_1$ is differentiable aswell). Is $S_2$ a regular surface?
My thoughts are that $S_2$ is also regular:
Let $S_1$ be a regular surface and $f:S_1 \rightarrow S_2$ a diffeomorphism. By definition of regular surface, for any $p\in S_1$ there exist a pair $(\varphi, U)$, where $U \subset \mathbb{R}^2$ is open and $\varphi:U \rightarrow S_1$ is an homeomorphism such that $p\in \varphi(U)$ and $d\varphi_p$ is injective.
The function $f \circ \varphi$ is a composition of a diffeomorphism with an homeomorphism, so it is an homeomorphism, and $d(f \circ \varphi)_p =d(f)_{\varphi(p)} \circ d(\varphi)_p$ is a composition of a bijective function (because f is diffeomorphism) with an injective function, so it is injective. Thus we conclude that $S_2$ is regular with parametrization $(f \circ \varphi, U)$.
However I haven't found this proposition in Do Carmo's book, which makes me think it is false. Is the claim true or have I made a mistake in my proof?