# Question 7706: Changed Codomain Must Contain All Original Outputs [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-0031) 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. R0014-07706 · q-20261012-advance-20261016-r07-surjective-0031 ## Original question and givens Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated). - G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated). ## 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=5;m=3;images=[0, 0, 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 original images=[0, 0, 0, 0, 1];candidate codomain=[0, 1, 2] After all original outputs eligible=Yes Conditions: none The new codomain may use arbitrary distinct integer labels,but must contain every original output.Do not discard outside outputs to manufacture function eligibility. Step identifiers and dependenciesRule: changed_codomain_full_range_guard · Uses: G001, S001 S003 · impliesBefore phase=changed codomain;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1] After fiber=[0, 1, 2, 3];missing=No Conditions: none For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty. Step identifiers and dependenciesRule: surjection_complete_target_fiber · Uses: G001, S002 S004 · impliesBefore phase=changed codomain;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1] After fiber=[4];missing=No Conditions: none For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty. Step identifiers and dependenciesRule: surjection_complete_target_fiber · Uses: G001, S003 S005 · impliesBefore phase=changed codomain;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1] After fiber=[];missing=Yes Conditions: none For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty. Step identifiers and dependenciesRule: surjection_complete_target_fiber · Uses: G001, S004 S006 · impliesBefore phase=changed codomain;targets=[0, 1, 2];fibers=[[0, 1, 2, 3], [4], []] After missing=[2];surjective=No Conditions: none Surjectivity is relative to the declared codomain:every target needs a preimage.An empty codomain has no targets to test and satisfies the condition;an empty domain with a nonempty codomain misses every target. Step identifiers and dependenciesRule: surjection_all_declared_targets · Uses: G001, S005 S007 · impliesBefore qualified=Yes;missing/status=[2] After surjectivity/status=No Conditions: none If an output is outside the new codomain,the specified function type is not valid,so surjectivity is unevaluated rather than False.When eligible,use all targets of the new codomain. Step identifiers and dependenciesRule: changed_codomain_guarded_result · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: b7ce9e0c5bd6548974bcff7ca3906606d3a3f5c7ded59e234e1dba471b7ae3c3 - candidate record SHA-256: 94a90b97b7eb0bb94854c7ea7037ed01067c3e8d652685c9e20a4325e0702def - semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77 - question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r07-surjective-0031.json)