# Question 5576: Actual Range and Unused Codomain

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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.

R0010-05576 · q-20261008-advance-20261012-r07-function-mapping-0026



## Original question and givens

The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?).


- G001 — The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire 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-function-mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/)





## Step-by-step derivation

S001 · impliesBefore

f:Fin(0)→Fin(0);source indices start at0

After

complete total map,not a partial program function

Conditions: none

Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal input.

Step identifiers and dependenciesRule: total_finite_mapping_scope · Uses: G001





S002 · impliesBefore

actual range=[];unused codomain=[]

After

range equals codomain=Yes

Conditions: none

Totality does not imply surjectivity;the range equals the codomain only if every target has a preimage.The empty map Fin(0)→Fin(0) satisfies this criterion.

Step identifiers and dependenciesRule: range_vs_declared_codomain · Uses: G001, S001









## Reverse checks


- 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

([], [], 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: 8674aa41437448a0f9dcf870eefbcef438e11dc9dad185ca81f1ceb427fb0769
- candidate record SHA-256: 190ff1b11703b5ad0fe0aac4290415c932e4d9339cbc1ffb554e531cc9d36639
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r07-function-mapping-0026.json)