Inverse of Relation Isomorphism is Relation Isomorphism

Theorem
Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures.

Let $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a bijection.

Then:
 * $\phi: \struct {S, \RR_1} \to \struct {T, \RR_2}$

is a relation isomorphism
 * $\phi^{-1}: \struct {T, \RR_2} \to \struct {S, \RR_1}$
 * $\phi^{-1}: \struct {T, \RR_2} \to \struct {S, \RR_1}$

is also a relation isomorphism.

Proof
Follows directly from the definition of relation isomorphism.