# Question 7731: Source Deletion with Declared Codomain Kept

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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.

R0014-07731 · q-20261012-advance-20261016-r07-surjective-0056



## Original question and givens

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).


- G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).





## 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=[2, 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

phase=original;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[2, 0, 0, 0, 1]

After

fiber=[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, S001





S003 · impliesBefore

phase=original;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[2, 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, S002





S004 · impliesBefore

phase=original;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[2, 0, 0, 0, 1]

After

fiber=[0];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=original;targets=[0, 1, 2];fibers=[[1, 2, 3], [4], [0]]

After

missing=[];surjective=Yes

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, S004





S006 · impliesBefore

deleted original input=2;old labels=[0, 1, 2, 3, 4]

After

kept labels=[0, 1, 3, 4];remaining images=[2, 0, 0, 1];codomain Fin(3)

Conditions: none

Delete the specified original input without changing the codomain or renumbering surviving labels.The remaining images list is ordered by retained labels,not a declaration of relabeled inputs.

Step identifiers and dependenciesRule: delete_input_keeps_original_labels_and_codomain · Uses: G001, S005





S007 · impliesBefore

phase=after deletion;target=0;source labels=[0, 1, 3, 4];images on labels=[2, 0, 0, 1]

After

fiber=[1, 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, S006





S008 · impliesBefore

phase=after deletion;target=1;source labels=[0, 1, 3, 4];images on labels=[2, 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, S007





S009 · impliesBefore

phase=after deletion;target=2;source labels=[0, 1, 3, 4];images on labels=[2, 0, 0, 1]

After

fiber=[0];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, S008





S010 · impliesBefore

phase=after deletion;targets=[0, 1, 2];fibers=[[1, 3], [4], [0]]

After

missing=[];surjective=Yes

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, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([2, 0, 0, 0, 1], [], [0, 1, 3, 4], [2, 0, 0, 1], [], Yes)

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: 4621f0eab4329d3bb4232eedbfa0ea83dc30089afafe72e395002253f2bd23a6
- candidate record SHA-256: 1b9605b48a4edd232efa46e7e5cc84ea64e6abf9c1b4f643fea5dd715d85928a
- 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-0056.json)