Inverse of Relation Isomorphism is Relation Isomorphism

Theorem
Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be relational structures.

Let $\phi: \left({S, \mathcal R_1}\right) \to \left({T, \mathcal R_2}\right)$ be a bijection.

Then:
 * $\phi: \left({S, \mathcal R_1}\right) \to \left({T, \mathcal R_2}\right)$

is a relation isomorphism iff:
 * $\phi^{-1}: \left({T, \mathcal R_2}\right) \to \left({S, \mathcal R_1}\right)$

is also a relation isomorphism.

Proof
Follows directly from the definition of relation isomorphism.