Riemann mapping theorem
Let $\Omega \subseteq \mathbb{C}$ be a region and $\mathbb{D}$ be the unit disk, $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. We define the equivalence relation of the set of the regions in $\mathbb{C}$ as follows: Two regions $\Omega_1,\Omega_2 \subseteq \mathbb{C}$ are conformally equivalent ($\Omega_1 \sim \Omega_2$) when there is $f\in H(\Omega_1)$ ($f$ analytic on $\Omega_1$) with $f:\Omega_1 \rightarrow \Omega_2$ is bijective (one to one).
Then we have: $\Omega \sim \mathbb{D} \Leftrightarrow \Omega$ is simply connected and $\Omega \neq \mathbb{C}$.
My questions revolves around the proof. Can someone describe me in descriptive steps (an outline in words) how the proof works?
Thank you very much for your time and patience, I appreciate it very much!