Order Isomorphism on Foundational Relation preserves Foundational Structure

Theorem

Let $A_1$ and $A_2$ be classes.

Let $\prec_1$ and $\prec_2$ be relations.

Let $\phi: \left({A_1, \prec_1}\right) \to \left({A_2, \prec_2}\right)$ be an order isomorphism.

Then $\left({A_1, \prec_1}\right)$ is a foundational structure iff $\left({A_2, \prec_2}\right)$ is also a foundational structure.

Proof

Take any nonempty subset $B \subseteq A$.

Then, $x$ is a minimal element of $B$ iff $\phi \left({ x }\right)$ is a minimal element of $\phi \left({ B }\right)$ by the fact that an Order Isomorphism Preserves Minimal Elements.

By the definition of foundational relation, $\left({ A_1 , \prec_1 }\right)$ is foundational if and only if $\left({ A_2 , \prec_2 }\right)$ is foundational.

$\blacksquare$