Like in Riemann's mapping theorem, we have a conformal mapping $f:\Omega\rightarrow\mathbb{D}$ (so $f$ is bijective and holomorphic), where $\mathbb{D}=\{|z|<1,\ z \in\mathbb{C}\}$ is the set of the open unit disks.
Why does it follow from Liouville's theorem, that $\Omega$ is not the whole $\mathbb{C}$, so $\Omega \neq \mathbb{C}$?
Liouville's theorem states that if $f$ is holomorphic on $\mathbb{C}$ and bounded, the $f$ is constant.
I really don't see how this determines $\Omega \neq \mathbb{C}$.
It would be so nice, if someone could walk me through this.
Thank you for your time,
Chris