What is the flaw in the answer for proving any rational function on PP^1 is constant? Le

What is the flaw in the answer for proving any rational function on ${\displaystyle P{P}^1}$ is constant?
Let ${\displaystyle \varphi :P{P}^1\to {\mathrm{\forall }}^1}$ be a rational function. Since ${\displaystyle {\mathrm{\forall }}^1\subset P{P}^1}$ we can think ${\displaystyle \varphi :P{P}^1\to P{P}^1}$ as rational function. As every rational map from ${\displaystyle P{P}^1\to P{P}^n}$ is regular, so ${\displaystyle \varphi }$ is regular. We also know that any regular function on ${\displaystyle P{P}^1}$ is constant, so ${\displaystyle \varphi }$ is constant.