# Equality on dependent function types

```agda
module foundation.equality-dependent-function-types where
```

<details><summary>Imports</summary>

```agda
open import foundation.dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.implicit-function-types
open import foundation.universe-levels

open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families
open import foundation-core.type-theoretic-principle-of-choice
```

</details>

## Idea

Given a family of types `B` over `A`, if we can characterize the
[identity types](foundation-core.identity-types.md) of each `B x`, then we can
characterize the identity types of `(x : A) → B x`.

## Properties

### Torsoriality

```agda
module _
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (x : A) → B x → UU l3}
  (is-torsorial-C : (x : A) → is-torsorial (C x))
  where

  is-torsorial-Eq-Π : is-torsorial (λ g → (x : A) → C x (g x))
  is-torsorial-Eq-Π =
    is-contr-equiv'
      ( (x : A) → Σ (B x) (C x))
      ( distributive-Π-Σ)
      ( is-contr-Π is-torsorial-C)

  is-torsorial-Eq-implicit-Π : is-torsorial (λ g → {x : A} → C x (g {x}))
  is-torsorial-Eq-implicit-Π =
    is-contr-equiv
      ( Σ ((x : A) → B x) (λ g → (x : A) → C x (g x)))
      ( equiv-Σ
        ( λ g → (x : A) → C x (g x))
        ( equiv-explicit-implicit-Π)
        ( λ _ → equiv-explicit-implicit-Π))
      ( is-torsorial-Eq-Π)
```

### Extensionality

```agda
module _
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2}
  (f : (x : A) → B x)
  (Eq-B : (x : A) → B x → UU l3)
  where

  map-extensionality-Π :
    ( (x : A) (y : B x) → (f x ＝ y) ≃ Eq-B x y) →
    ( g : (x : A) → B x) → f ＝ g → ((x : A) → Eq-B x (g x))
  map-extensionality-Π e .f refl x = map-equiv (e x (f x)) refl

  abstract
    is-equiv-map-extensionality-Π :
      (e : (x : A) (y : B x) → (f x ＝ y) ≃ Eq-B x y) →
      (g : (x : A) → B x) → is-equiv (map-extensionality-Π e g)
    is-equiv-map-extensionality-Π e =
      fundamental-theorem-id
        ( is-torsorial-Eq-Π
          ( λ x →
            fundamental-theorem-id'
              ( λ y → map-equiv (e x y))
              ( λ y → is-equiv-map-equiv (e x y))))
        ( map-extensionality-Π e)

  extensionality-Π :
    ( (x : A) (y : B x) → (f x ＝ y) ≃ Eq-B x y) →
    ( g : (x : A) → B x) → (f ＝ g) ≃ ((x : A) → Eq-B x (g x))
  pr1 (extensionality-Π e g) = map-extensionality-Π e g
  pr2 (extensionality-Π e g) = is-equiv-map-extensionality-Π e g
```

## See also

- Equality proofs in the fiber of a map are characterized in
  [`foundation.equality-fibers-of-maps`](foundation.equality-fibers-of-maps.md).
- Equality proofs in cartesian product types are characterized in
  [`foundation.equality-cartesian-product-types`](foundation.equality-cartesian-product-types.md).
- Equality proofs in dependent pair types are characterized in
  [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md).
- Function extensionality is postulated in
  [`foundation.function-extensionality`](foundation.function-extensionality.md).