Group Isomorphism Preserves Identity/Proof 1

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

Proof
An group isomorphism is by definition a group epimorphism.

The result follows from Epimorphism Preserves Identity and Epimorphism Preserves Inverses.