If $\Omega = \emptyset$, the claim holds vacuously so assume that $\Omega$ is not empty and fix $z_0, z_1 \in \Omega$. Since $\Omega$ is connected, it is in particular path-connected, whence there exists an

D5023: Oriented complex curve $\gamma$ which starts at $z_0$ and ends at $z_1$. Since $f$ is a

D5005: Complex function primitive for $f'$, then applying result

R1558: Second fundamental theorem of complex integral calculus, we have
\begin{equation}
0
= \int_{\gamma} 0 \, d s
= \int_{\gamma} f'(s) \, d s
= f(z_1) - f(z_0)
\end{equation}
Adding $f(z_0)$ to both sides, we find $f(z_1) = f(z_0)$. Since $z_0, z_1 \in \Omega$ were arbitrary, it follows that $f$ is constant on $\Omega$. $\square$