DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Extensional relations and collapse maps

Definition

A setlike relation $R$ on $X$ is extensional when ${pred}_{R} (x) = {pred}_{R} (y)$ implies $x = y$ for $x, y \in X$ . If $R$ is also well-founded, its collapse map is the unique definable function

$π (x) = {π (y) : y R x} .$

Existence and uniqueness follow from well-founded recursion with $G (x, h) = ran (h)$ , a set by Replacement. A collapse map is defined even without extensionality; injectivity is a further conclusion requiring extensionality. The construction for a supplied well-founded relation uses no ambient Foundation.

Conventions and prerequisites: Recursion on well-founded setlike relations.

Depends on

Recursion on well-founded setlike relations

Used by

Well-founded pointed graphs have unique decorations Corollary
Extensionality is needed for injective collapse Counterexample
An extensional collapse is injective Lemma
Mostowski collapse for extensional relations Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marks, Set Theory, Berkeley edition — 6.10–6.11 pp.31–32. (standard reference, not scraped)