Suppose that $f=u+iv$ is holomorphic in an open set $\Omega$. Prove that $f$ is constant if $f$ is constant.

My attempt:

If $f\equiv 0$ then its trivial. If $|f|=c$ then $\partial\_x|f|=\partial\_y|f|=0$ so $\dfrac{u\partial\_xu+v\partial\_xv}{\sqrt{u^2+v^2}}=\dfrac{u\partial\_yu+v\partial\_yv}{\sqrt{u^2+v^2}}=0$ which implies that $\begin{cases}u\partial\_xu+v\partial\_xv=0\\ u\partial\_yu+v\partial\_yv=0 \end{cases}$ (since $\sqrt{u^2+v^2}=|f|$ is constant). But can't seem to be able to prove that $\partial\_x u,\partial \_x v=0$ (and thus prove $f'(z)=0$). What could I try next? Im kind of stuck.