Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Gamma \subset \mathbb{C}$ be a curve which has no self intersections. If we assume that for any $\omega \in \Gamma$, $f^{-1}(\{\omega\}) \ne \emptyset$, then my question is that, can we find a curve $\gamma \subset \mathbb{C}$ such that $f$ is one-to-one on $\gamma$ and that $f(\gamma)=\Gamma$?

I believe this is true because such function $f$ is locally one-to-one, and even if points on $\Gamma$ can be possibly taken infinitely many times on the complex plane $\mathbb{C}$, we can always choose a certain "branch" such that the restricition of $f$ on such a "branch" is one-to-one. But I'm not sure how to give a rigorous argument to validate the existence of such "branch". Any comments or suggestions will be fully appreciated.