Order Isomorphism on Strictly Well-Founded Relation preserves Strictly Well-Founded Structure

Theorem
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
Follows immediately from the fact that Order Isomorphism Preserves Minimal Elements and the definition of foundational relations.