Identity Mapping is Relation Isomorphism

Theorem
Let $\struct {S, \RR}$ be a relational structure.

Then the identity mapping $I_S: S \to S$ is a relation isomorphism from $\struct {S, \RR}$ to itself.

Proof
By definition of identity mapping:
 * $\forall x \in S: \map {I_S} x = x$

So:
 * $x \mathrel \RR y \implies \map {I_S} x \mathrel \RR \map {I_S} y$

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

Hence:
 * $\map {I_S^{-1} } x = x$

So:
 * $x \mathrel \RR y \implies \map {I_S^{-1} } x \mathrel \RR \map {I_S^{-1} } y$