Group Epimorphism is Isomorphism iff Kernel is Trivial/Proof 2

Proof
From Kernel is Trivial iff Monomorphism, $\phi$ is a monomorphism $K = \left\{{e_G}\right\}$.

By definition, a group $G$ is an epimorphism is an isomorphism $G$ is also a monomorphism.

Hence the result.