Group Isomorphism Preserves Inverses/Proof 2

Proof
Let $g \in G$.

It follows from Inverse in Group is Unique that $\phi \left({g^{-1} }\right)$ is the unique inverse element of $\phi \left({g}\right)$ in $\left({H, *}\right)$.

That is:
 * $\forall g \in G: \phi \left({g^{-1}}\right) = \left({\phi \left({g}\right)}\right)^{-1}$