Group Isomorphism Preserves Inverses/Proof 2

Proof
Let $g \in G$.

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

That is:
 * $\forall g \in G: \map \phi {g^{-1} } = \paren {\map \phi g}^{-1}$