Editing Mostowski collapse lemma (section)

==Generalizations==
Every well-founded set-like relation can be embedded into a well-founded set-like extensional relation. This implies the following variant of the Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.

A mapping ''F'' such that ''F''(''x'') = {''F''(''y'') : ''y R x''} for all ''x'' in ''X'' can be defined for any well-founded set-like relation ''R'' on ''X'' by [[well-founded relation|well-founded recursion]]. It provides a [[homomorphism#Relational structures|homomorphism]] of ''R'' onto a (non-unique, in general) transitive class. The homomorphism ''F'' is an isomorphism if and only if ''R'' is extensional.

The well-foundedness assumption of the Mostowski lemma can be alleviated or dropped in [[non-well-founded set theory|non-well-founded set theories]]. In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class. In set theory with [[Aczel's anti-foundation axiom]], every set-like relation is [[bisimulation|bisimilar]] to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.