## Range of biholomorphic functions in $\mathbb{C}$ [closed]

How could one prove that among the biholomorphic functions that map a fixed simply-connected open domain $D$ into open subsets of $\mathbb{C}$, there is such a biholomorphic map $f$ for each open simply-connected $U\subset\mathbb{C}$, such that $f(D)=U$? Also I would like to avoid the use of Riemann mapping theorem in such a proof.

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I would think that you are exactly asking about the Riemann mapping theorem, so it is not clear to me what part is to be avoided, or what purpose it might serve to do so. – paul garrett Jul 15 2011 at 20:23