# Function mapping

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The notation f:X→Y states that the function f maps the domain X into the codomain Y.

mko-function-mapping · definition · v0.8.0



## Mathematical statement

A function f:X→Y assigns exactly one output in Y to every input in X; the codomain Y need not equal the function's actual range.





## Representative formula (exact TeX)

f:X\to Y





## Assumptions


- X and Y are specified sets or compatible types.
- Every input in X has exactly one output under f.





## Symbol scope


- f — function
- X — domain
- Y — codomain
- → — mapping arrow





## Intuition, definition, and misconceptions

The core of a function is not a drawn curve but a rule that associates every permitted input with exactly one output.

The declaration f:X→Y specifies the domain and codomain. The set of outputs actually attained may be only a subset of Y.

The Lean theorem function_total_unique proves, for every f:X→Y and x:X, that there exists a unique y:Y satisfying f x=y.

A common mistake is to treat the codomain, range, and graph of a function as the same object, or to identify effectful or failure-prone program functions directly with pure total mathematical functions.





## Adopted derivation guide

[Function mapping: deciding whether a rule is a function, and separating codomain, range and graph](https://ocme.evemisslab.com/en/derivations/derivation-mko-function-mapping-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) — Function inputs and outputs belong to the domain and codomain respectively.





## Verification boundary

Lean formalizes the core total-function claim that every input has a unique output. Partial functions, side effects, randomness, and execution failure lie outside this statement.

formalized_total_function_core


- [evidence-sha256-d999133c1ae711b73752af8ee4b1b3ce74ca135ff7c06aabbe9fc8a3186f434b](https://ocme.evemisslab.com/data/evidence/d999133c1ae711b73752af8ee4b1b3ce74ca135ff7c06aabbe9fc8a3186f434b.json)





## Next obligation

Add objects for range, injection, surjection, bijection, and partial functions, keeping the typed codomain distinct from the set of outputs actually attained.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/function-mapping.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-function-mapping/index.md)