PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

A morphism carries members to members and preserves equivalence

Statement

Let $f : A \to B$ be a morphism.

If $x : X \to A$ is a member of $A$ , then $f x : X \to B$ is a member of $B$ .
If $x \equiv y$ as members of $A$ , then $f x \equiv f y$ as members of $B$ .

Facts & Assumptions

Given: A morphism $f : A \to B$ and members $x : X \to A$ , $y : Y \to A$ .

[L1]

Member equivalence means equality after precomposition by one common pair of epimorphisms (Equivalence of members).

Proof

technique · direct

1.1

The composite $f x : X \to B$ is again a morphism into $B$ , so it is a member of $B$ .

given

1.2

If $x \equiv y$ , choose epimorphisms $u : W \to X$ and $v : W \to Y$ with $x u = y v$ by [L1]. Postcomposing with $f$ gives $f x u = f y v$ , and the same epimorphisms witness $f x \equiv f y$ .

L1algebra

2.1

Therefore every morphism carries members to members and preserves their equivalence.

step 1.1step 1.2∎

Depends on

Equivalence of members

Used by

Two morphisms agreeing on every member need not be equal Counterexample
A zero arrow is detected by members Theorem
Epimorphy is detected by members Theorem
Exactness is detected by members Theorem
Monicity is detected by members Theorem
The subtraction surrogate Theorem

Dependency tree · two levels

3 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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)