Isomorphism Preserves Inverses/Proof 1

Proof
Let $\struct {S, \circ}$ be an algebraic structure in which $\circ$ has an identity $e_S$.

From Epimorphism Preserves Identity, it follows that $\struct {T, *}$ also has an identity, which is $\map \phi {e_S}$.

Let $y$ be an inverse of $x$ in $\struct {S, \circ}$.

Then:

So $\map \phi y$ is an inverse of $\map \phi x$ in $\struct {T, *}$.

As $\phi$ is an isomorphism, it follows from Inverse of Algebraic Structure Isomorphism is Isomorphism that $\phi^{-1}$ is also a isomorphism.

Thus the result for $\map \phi x$ can be applied to $\map {\phi^{-1} } {\map \phi x}$.