This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

The univalence axiom implies function extensionality

module foundation.univalence-implies-function-extensionality where
Imports 
open import foundation.dependent-pair-types
open import foundation.equivalence-induction
open import foundation.function-extensionality
open import foundation.postcomposition-functions
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels
open import foundation.weak-function-extensionality

open import foundation-core.contractible-maps
open import foundation-core.contractible-types
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications

Idea

The univalence axiom implies function extensionality.

Theorem

abstract
  weak-funext-univalence : {l : Level} → weak-function-extensionality-Level l l
  weak-funext-univalence A B is-contr-B =
    is-contr-retract-of
      ( fiber (postcomp A pr1) id)
      ( ( λ f → ((λ x → (x , f x)) , refl)) ,
        ( λ h x → tr B (htpy-eq (pr2 h) x) (pr2 (pr1 h x))) ,
        ( refl-htpy))
      ( is-contr-map-is-equiv
        ( is-equiv-postcomp-univalence A (equiv-pr1 is-contr-B))
        ( id))

abstract
  funext-univalence :
    {l : Level} → function-extensionality-Level l l
  funext-univalence f = funext-weak-funext weak-funext-univalence f