Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a map. Suppose that $F$ is bijective and that $F$ is a local diffeomorphism: for every $p \in M$, there exist an open neighbourhood $U_p \subset M$ of $p$ and an open neighbourhood $V_p \subset N$ of $F(p)$ such that $F(U_p) = V_p$ and the restriction
paragraph
admin
\begin{align*}
F|_{U_p}: U_p \to V_p
\end{align*}
latex_env
admin
is a diffeomorphism. Then $F: M \to N$ is a diffeomorphism.