Definition df-idfu 17787

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

Step	Hyp	Ref	Expression
1		cidfu 17783	. 2 class idfunc
2		vt	. . 3 setvar 𝑡
3		ccat 17591	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1541	. . . . 5 class 𝑡
6		cbs 17140	. . . . 5 class Base
7	5, 6	cfv 6493	. . . 4 class (Base‘𝑡)
8		cid 5519	. . . . . 6 class I
9	4	cv 1541	. . . . . 6 class 𝑏
10	8, 9	cres 5627	. . . . 5 class ( I ↾ 𝑏)
11		vz	. . . . . 6 setvar 𝑧
12	9, 9	cxp 5623	. . . . . 6 class (𝑏 × 𝑏)
13	11	cv 1541	. . . . . . . 8 class 𝑧
14		chom 17192	. . . . . . . . 9 class Hom
15	5, 14	cfv 6493	. . . . . . . 8 class (Hom ‘𝑡)
16	13, 15	cfv 6493	. . . . . . 7 class ((Hom ‘𝑡)‘𝑧)
17	8, 16	cres 5627	. . . . . 6 class ( I ↾ ((Hom ‘𝑡)‘𝑧))
18	11, 12, 17	cmpt 5180	. . . . 5 class (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))
19	10, 18	cop 4587	. . . 4 class ⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩
20	4, 7, 19	csb 3850	. . 3 class ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩
21	2, 3, 20	cmpt 5180	. 2 class (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
22	1, 21	wceq 1542	1 wff idfunc = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)

This definition is referenced by:  idfuval  17804  idfurcl  49379