Group Isomorphism Preserves Identity/Proof 2

Theorem
Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group isomorphism.

Let:
 * $e_G$ be the identity of $\left({G, \circ}\right)$
 * $e_H$ be the identity of $\left({H, *}\right)$.

Then:
 * $\phi \left({e_G}\right) = e_H$
 * $\forall g \in G: \phi \left({g^{-1}}\right) = \left({\phi \left({g}\right)}\right)^{-1}$

Preservation of Identity
It follows from Identity is only Idempotent Element in Group that $\phi \left({e_G}\right)$ is the identity of $H$.

That is, $\phi \left({e_G}\right) = e_H$.

Preservation of Inverses
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:
 * $\forall g \in G: \phi \left({g^{-1}}\right) = \left({\phi \left({g}\right)}\right)^{-1}$