Homomorphism to Group Preserves Inverses

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\left({T, *}\right)$ be a group.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a homomorphism.

Let $\left({S, \circ}\right)$ have an identity $e_S$.

Let $x^{-1}$ be an inverse of $x$ for $\circ$.

Then $\phi \left({x^{-1}}\right)$ is an inverse of $\phi \left({x}\right)$ for $*$.

Proof
By hypothesis, $\left({T, *}\right)$ is a group.

By group axiom $G2$, $\left({T, *}\right)$ has an identity.

Thus Homomorphism with Identity Preserves Inverses can be applied.