# The univalence axiom implies function extensionality

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-02-16.

module foundation.univalence-implies-function-extensionality where

Imports
open import foundation.dependent-pair-types
open import foundation.equivalence-induction
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-extensionality
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} {A : UU l} {B : A → UU l} → weak-function-extensionality A B
weak-funext-univalence {A = A} {B} is-contr-B =
is-contr-retract-of
( fiber (postcomp A (pr1 {B = B})) id)
( pair
( λ f → pair (λ x → pair x (f x)) refl)
( pair
( 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} {A : UU l} {B : A → UU l} (f : (x : A) → B x) →
function-extensionality f
funext-univalence {A = A} {B} f =
funext-weak-funext (λ A B → weak-funext-univalence) A B f