Inverse Relational Structures of Isomorphic Structures are Isomorphic

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

Let ${\RR_1}^{-1}$ and ${\RR_2}^{-1}$ be the inverses of $\RR_1$ and $\RR_2$ respectively.

Let $f: \struct {S, \RR_1} \to \struct {T, \RR_2}$ be a relation isomorphism.

Then $f: \struct {S, {\RR_1}^{-1} } \to \struct {T, {\RR_2}^{-1} } $ is also a relation isomorphism.