Inverse of Algebraic Structure Isomorphism is Isomorphism

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

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

Then $\phi$ is an isomorphism iff $\phi^{-1}: \left({T, *}\right) \to \left({S, \circ}\right)$ is also an isomorphism.

Proof
As $\phi$ is a bijection, then $\exists \phi^{-1}$ such that $\phi^{-1}$ is also a bijection from Bijection iff Inverse is Bijection. That is:


 * $\exists \phi^{-1}: \left({T, *}\right) \to \left({S, \circ}\right)$

It follows that:

So $\phi^{-1}: \left({T, *}\right) \to \left({S, \circ}\right)$ is a homomorphism, and (from above) bijective, and thus an isomorphism.

Applying the same result in reverse, we have that if $\phi^{-1}: \left({T, *}\right) \to \left({S, \circ}\right)$ is an isomorphism, then $\left({\phi^{-1}}\right)^{-1}: \left({S, \circ}\right) \to \left({T, *}\right)$ is an isomorphism.

But by Inverse of Inverse of Bijection, $\left({\phi^{-1}}\right)^{-1} = \phi$ and hence the result.