On Rudin's proof of Picard's little theorem

Question

I got a question in the proof of picard little theorem by W. Rudin in the textbook "Real & Complex Analysis(third edition)". 16.19 Theorem tries to show that $\phi$ in modular group satisfies $\phi(W)\cap W=\emptyset$ where W is some fundamental domain in $\mathbb{C}$. To get a contradiction, it is first shown that $$\text{Im }\phi(z)=\text{Im }z/|cz+d|^2<\text{Im }z\tag 1$$ since $|cz+d|>1$. It then says using similar argument to get $\text{Im}(z)=\text{Im }\phi^{-1}(\phi(z))<\text{Im }\phi(z)$. However, I only know that $$\text{Im }\phi^{-1}(z)=\text{Im }z/|cz-a|^2\tag 2$$ which does not imply $\text{Im }\phi^{-1}(z)<\text{Im }z$ for $z \in \phi(W)$. Indeed, if we replace z by a $\phi(w)$ for some $w\in W$, equation (2) is the same to the equation (1). Can anyone help with that?