Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

Question

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$, but I'm not sure how to proceed. Can someone please help