Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the pigeonhole principle there exist $u, u' \in \mathbb{R}^2$ where $u\neq u'$ which map to the same element $\phi(u)$. But then $\phi$ is not an isometry since $d(u,u') > d(\phi(u),\phi(u))=0$.
My issue is with using pigeonhole principle for uncountable sets, which feels flawed to me.