Group Homomorphism Preserves Inverses/Proof 1

Proof
Let $x \in G$.

Then:

So, by definition, $\map \phi {x^{-1} }$ is the right inverse of $\map \phi x$.

Similarly:

So, again by definition, $\map \phi {x^{-1} }$ is the left inverse of $\map \phi x$.

Finally, as $\map \phi {x^{-1} }$ is both:
 * a left inverse of $\map \phi x$

and:
 * a right inverse of $\map \phi x$

it is by definition an inverse.

From Inverse in Group is Unique, $\map \phi {x^{-1} }$ is the only such element.

Hence the result.