Identity Mapping is Relation Isomorphism

Theorem
Let $\left({S, \mathcal R}\right)$ be a relational structure.

Then the identity mapping $I_S: S \to S$ is an relation isomorphism from $\left({S, \mathcal R}\right)$ to itself.

Proof
By definition of identity mapping:
 * $\forall x \in S: I_S \left({x}\right) = x$

So:
 * $x \mathop{\mathcal R} y \implies I_S \left({x}\right) \mathop{\mathcal R} I_S \left({y}\right)$

From Identity Mapping is Bijection, $I_S$ is a bijection.

Hence:
 * $I_S^{-1} \left({x}\right) = x$

So:
 * $x \mathop{\mathcal R} y \implies I_S^{-1} \left({x}\right) \mathop{\mathcal R} I_S^{-1} \left({y}\right)$