Isomorphism Preserves Inverses

Theorem
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Then $x^{-1}$ is an inverse of $x$ for $\circ$ $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.

That is, :
 * $\map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$