Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping

Theorem
Let $( A, \prec )$ be a strict well-ordering. Let $A$ also be a set.

Then, there is a unique ordinal $x$ and unique mapping $f$ such that $f : x \to A$ is an order isomorphism.

Proof
We can prove the existence of $x$ and $f$ from Every Woset is Isomorphic to a Unique Ordinal.

The uniqueness of $x$ also follows from Every Woset is Isomorphic to a Unique Ordinal.

The uniqueness of $f$ follows from Order Isomorphism Between Wosets is Unique.