Group Isomorphism Preserves Inverses

Theorem
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism.

Let:
 * $e_G$ be the identity of $\struct {G, \circ}$
 * $e_H$ be the identity of $\struct {H, *}$.

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