PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

The coimage projection is epic and the image inclusion is monic

Statement

For every morphism $f$ in a category with kernels and cokernels, the defining map $q_{f} : A \to coim (f)$ is epic and the defining map $i_{f} : im (f) \to B$ is monic.

Facts & Assumptions

Given: A morphism $f : A \to B$ with coimage projection $q_{f}$ and image inclusion $i_{f}$ .

[L1]

The coimage is defined as a cokernel and the image as a kernel (Image and coimage in a category with kernels and cokernels).

[L2]

Every coequalizer is epic and every equalizer is monic (Every equalizer is a monomorphism, and every coequalizer is an epimorphism).

Proof

technique · direct

1.1

By [L1], the map $q_{f}$ is a cokernel of a kernel of $f$ . Therefore [L2] makes $q_{f}$ epic.

L1L2

2.1

Again by [L1], the map $i_{f}$ is a kernel of a cokernel of $f$ . Therefore [L2] makes $i_{f}$ monic.

L1L2∎

Depends on

Used by

Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism Theorem

Dependency tree · two levels

5 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

The Stacks Project, Section 12.3, Fact 12.3.11 (standard reference, not scraped)