 Solution

The Mostowski collapse theorem states that if $R$ is a well-founded extensional relation on a set $X$, then there is a unique transitive set $M$ and a unique bijection $\pi :X\to M$ such thatExtensionality means that distinct elements of $X$ have distinct predecessor sets. The collapse is characterized by the recursionWell-founded recursion defines it, extensionality makes it injective, and its range is transitive. Uniqueness follows by well-founded induction. For a class version, the relation must also be set-like; its predecessor sets must be sets.