# Question 7779: Two Target-Type Guards, Complete Fibers, and Least Preimages [Return to catalogue page 39](https://ocme.evemisslab.com/en/questions/page/39/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r07-surjective-foundations/#q-20261012-advance-20261016-r07-surjective-0079) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0014-07779 · q-20261012-advance-20261016-r07-surjective-0079 ## Original question and givens Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain). - G001 — Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-surjective](https://ocme.evemisslab.com/en/mko/mko-surjective/) ## Step-by-step derivation S001 · impliesBefore n=3;m=2;images=[0, 0, 1] After all original inputs and outputs form a typed total table;labels start at0 Conditions: none First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity. Step identifiers and dependenciesRule: finite_surjection_typed_scope · Uses: G001 S002 · impliesBefore query0=0;codomain Fin(2);original images=[0, 0, 1] After eligible=Yes;fiber/status=[0, 1];least/status=0 Conditions: none First check0≤target