Injective smooth map which is immersive and submersive is actually a global diffeomorphism

Question

Let $U\subset \Bbb R^n$ be an open subset and $f:U\to \Bbb R^n$ be an injective smooth map where $Df_p$ is invertible for each $p\in U$. I want to show that $f$ is a global diffeomorphism from $U$ to $f(U)$.

The inverse value theorem says that for each $p\in U$ I can find $U_a\subseteq U$ and $V_a\subseteq \Bbb R^n$ such that $f|_{U_a}:U_a\to V_a$ is a diffeomorphism, with smooth inverse I'll denote $g_a$. Then I want to 'glue' the $g_a$ together to get a smooth (global) inverse to $f$. I'm not immediately sure how to make this rigorous.

I think the argument should be like: Since $F$ is injective, for any $a,b\in U$ such that $V_a\cap V_b\ne \emptyset$ $g_a|_{V_a\cap V_b}=g_b|_{V_a\cap V_b}$, in which case $g:f(U)=\bigcup_{a\in U}V_a\to U$ given by $g(x)=g_a(x)$ on $V_a$ is well defined, and smooth.

Answer 1

We know that $f(U) \subseteq \mathbb{R}^n$ is an open subset since $Df_p$ is invertible for each $p \in U$. So $f : U \to f(U)$ is a bijective local diffeomorphism.

To show $f$ is diffeomorphism, we only need to verify that $f$ and $f^{-1}$ are smooth.

One can show that any map $F : M \to N$, $M \subseteq \mathbb{R}^m$ and $N \subseteq \mathbb{R}^n$ is smooth iff every point $p \in M$ has a neighbourhood $U$ such that $F|_U : U \to N$ is smooth.

Now because $f$ is a local diffeomorphism, we always has a neighbourhood $V \subseteq U$ of each point such that $f|_V : V \to f(V)\subseteq f(U)$ diffeomorphism. The inclusion map $\iota : f(V) \hookrightarrow f(U)$ is smooth. Hence the composition $f|_V =f \circ \iota : V \to f(U)$ is smooth. So $f : U \to f(U)$ smooth. By similar manner $f^{-1} : f(U) \to U$ is also smooth