Definition df-cid 17563

Description: Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)

Assertion

Ref	Expression
df-cid	⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))

Distinct variable group:   𝑏,𝑐,𝑓,𝑔,ℎ,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-cid

Step	Hyp	Ref	Expression
1		ccid 17559	. 2 class Id
2		vc	. . 3 setvar 𝑐
3		ccat 17558	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1540	. . . . 5 class 𝑐
6		cbs 17094	. . . . 5 class Base
7	5, 6	cfv 6501	. . . 4 class (Base‘𝑐)
8		vh	. . . . 5 setvar ℎ
9		chom 17158	. . . . . 6 class Hom
10	5, 9	cfv 6501	. . . . 5 class (Hom ‘𝑐)
11		vo	. . . . . 6 setvar 𝑜
12		cco 17159	. . . . . . 7 class comp
13	5, 12	cfv 6501	. . . . . 6 class (comp‘𝑐)
14		vx	. . . . . . 7 setvar 𝑥
15	4	cv 1540	. . . . . . 7 class 𝑏
16		vg	. . . . . . . . . . . . . 14 setvar 𝑔
17	16	cv 1540	. . . . . . . . . . . . 13 class 𝑔
18		vf	. . . . . . . . . . . . . 14 setvar 𝑓
19	18	cv 1540	. . . . . . . . . . . . 13 class 𝑓
20		vy	. . . . . . . . . . . . . . . 16 setvar 𝑦
21	20	cv 1540	. . . . . . . . . . . . . . 15 class 𝑦
22	14	cv 1540	. . . . . . . . . . . . . . 15 class 𝑥
23	21, 22	cop 4597	. . . . . . . . . . . . . 14 class ⟨𝑦, 𝑥⟩
24	11	cv 1540	. . . . . . . . . . . . . 14 class 𝑜
25	23, 22, 24	co 7362	. . . . . . . . . . . . 13 class (⟨𝑦, 𝑥⟩𝑜𝑥)
26	17, 19, 25	co 7362	. . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓)
27	26, 19	wceq 1541	. . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
28	8	cv 1540	. . . . . . . . . . . 12 class ℎ
29	21, 22, 28	co 7362	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
30	27, 18, 29	wral 3060	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
31	22, 22	cop 4597	. . . . . . . . . . . . . 14 class ⟨𝑥, 𝑥⟩
32	31, 21, 24	co 7362	. . . . . . . . . . . . 13 class (⟨𝑥, 𝑥⟩𝑜𝑦)
33	19, 17, 32	co 7362	. . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔)
34	33, 19	wceq 1541	. . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
35	22, 21, 28	co 7362	. . . . . . . . . . 11 class (𝑥ℎ𝑦)
36	34, 18, 35	wral 3060	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
37	30, 36	wa 396	. . . . . . . . 9 wff (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
38	37, 20, 15	wral 3060	. . . . . . . 8 wff ∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
39	22, 22, 28	co 7362	. . . . . . . 8 class (𝑥ℎ𝑥)
40	38, 16, 39	crio 7317	. . . . . . 7 class (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))
41	14, 15, 40	cmpt 5193	. . . . . 6 class (𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)))
42	11, 13, 41	csb 3858	. . . . 5 class ⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)))
43	8, 10, 42	csb 3858	. . . 4 class ⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)))
44	4, 7, 43	csb 3858	. . 3 class ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)))
45	2, 3, 44	cmpt 5193	. 2 class (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))
46	1, 45	wceq 1541	1 wff Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))

This definition is referenced by:  cidfval  17570  cidffn  17572