Group Epimorphism is Isomorphism iff Kernel is Trivial/Proof 2

Theorem
Let $\left({G, \oplus}\right)$ and $\left({H, \odot}\right)$ be groups.

Let $\phi: \left({G, \oplus}\right) \to \left({H, \odot}\right)$ be a group epimorphism.

Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively.

Let $K = \ker \left({\phi}\right)$ be the kernel of $\phi$.

Then:
 * the epimorphism $\phi$ is an isomorphism


 * $K = \left\{{e_G}\right\}$

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.