Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-idfu Structured version   Visualization version   GIF version

Definition df-idfu 16741
 Description: Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
df-idfu idfunc = (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
Distinct variable group:   𝑡,𝑏,𝑧

Detailed syntax breakdown of Definition df-idfu
StepHypRef Expression
1 cidfu 16737 . 2 class idfunc
2 vt . . 3 setvar 𝑡
3 ccat 16547 . . 3 class Cat
4 vb . . . 4 setvar 𝑏
52cv 1631 . . . . 5 class 𝑡
6 cbs 16080 . . . . 5 class Base
75, 6cfv 6050 . . . 4 class (Base‘𝑡)
8 cid 5174 . . . . . 6 class I
94cv 1631 . . . . . 6 class 𝑏
108, 9cres 5269 . . . . 5 class ( I ↾ 𝑏)
11 vz . . . . . 6 setvar 𝑧
129, 9cxp 5265 . . . . . 6 class (𝑏 × 𝑏)
1311cv 1631 . . . . . . . 8 class 𝑧
14 chom 16175 . . . . . . . . 9 class Hom
155, 14cfv 6050 . . . . . . . 8 class (Hom ‘𝑡)
1613, 15cfv 6050 . . . . . . 7 class ((Hom ‘𝑡)‘𝑧)
178, 16cres 5269 . . . . . 6 class ( I ↾ ((Hom ‘𝑡)‘𝑧))
1811, 12, 17cmpt 4882 . . . . 5 class (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))
1910, 18cop 4328 . . . 4 class ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩
204, 7, 19csb 3675 . . 3 class (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩
212, 3, 20cmpt 4882 . 2 class (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
221, 21wceq 1632 1 wff idfunc = (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
 Colors of variables: wff setvar class This definition is referenced by:  idfuval  16758
 Copyright terms: Public domain W3C validator