Equivalent condition for an extension of local fields to be unramified

Question

Let $L/K$ be a degree $n$ finite Galois extension of local fields with corresponding residue field extension $\ell/k$ having residue class degree $f$ and ramification index $e$. We have a canonical map $\pi:\operatorname{Gal}(L/K)\to\operatorname{Gal}(\ell/k)$ given by restricting an automorphism $\sigma$ to the respective valuation rings then factoring out the maximal ideal of $\mathcal{O}_L$. I have seen it claimed that $L/K$ is unramified iff $\pi$ is an isomorphism. One direction is clear from the equation $ef=n$: if $\pi$ is an isomorphism, then $f=n$ and thus $e=1$. However, it is not obvious to me how to prove the other direction. I've only seen a proof that $\pi$ is injective in the case that the extension is generated by roots of unity having order prime to the residue characteristic of $K$. Does unramified automatically imply injectivity or surjectivity in some simple way, or is there a more subtle argument?