Group Homomorphism Preserves Inverses/Proof 1

Proof
Let $x \in G$.

Then:

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

Similarly:

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

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

and:
 * a right inverse of $\phi \left({x}\right)$

it is by definition an inverse.

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

Hence the result.