Relation Isomorphism Preserves Antisymmetry

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

Let $\left({S, \mathcal R_1}\right)$ and $\left({T, \mathcal R_2}\right)$ be (relationally) isomorphic.

Then $\mathcal R_1$ is an antisymmetric relation $\mathcal R_2$ is also an antisymmetric relation.

Proof
Let $\phi: S \to T$ be a relation isomorphism.

By Inverse of Relation Isomorphism is Relation Isomorphism it follows that $\phi^{-1}: T \to S$ is also a relation isomorphism.

Thus WLOG it suffices to prove only that if $\mathcal R_1$ is antisymmetric, then also $\mathcal R_2$ is antisymmetric.

So, suppose $\mathcal R_1$ is an antisymmetric relation.

Let $y_1, y_2 \in T$ such that both $y_1 \mathrel {\mathcal R_2}, y_2$ and $y_2 \mathrel {\mathcal R_2} y_1$.

Let $x_1 = \phi^{-1} \left({y_1}\right)$ and $x_2 = \phi^{-1} \left({y_2}\right)$.

As $\phi$ is a bijection it follows from Inverse Element of Bijection that $y_1 = \phi \left({x_1}\right)$ and $y_2 = \phi \left({x_2}\right)$.

As $\phi^{-1}$ is a relation isomorphism it follows that:
 * $x_1 = \phi^{-1} \left({y_1}\right) \mathrel {\mathcal R_1} \phi^{-1} \left({y_2}\right) = x_2$
 * $x_2 = \phi^{-1} \left({y_2}\right) \mathrel {\mathcal R_1} \phi^{-1} \left({y_1}\right) = x_1$

As $\mathcal R_1$ is an antisymmetric relation it follows that $x_1 = x_2$.

As $\phi$ is a bijection it follows that:
 * $y_1 = \phi \left({x_1}\right) = \phi \left({x_2}\right) = y_2$

Hence $y_1 = y_2$ and so by definition, $\mathcal R_2$ is an antisymmetric relation.

Hence the result.