Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be holomorphic, and let $z_0\in\Omega$ satisfy $f'(z_0)\ne0$. Then there exist open neighbourhoods $U\subset\Omega$ of $z_0$ and $V\subset\mathbb C$ of $f(z_0)$ such that $f|_U:U\to V$ is bijective, $(f|_U)^{-1}:V\to U$ is holomorphic, and
\begin{align*}
\left((f|_U)^{-1}\right)'(w)=\frac{1}{f'((f|_U)^{-1}(w))}
\end{align*}
for every $w\in V$.