Definition df-cid 17606

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 17602	. 2 class Id
2		vc	. . 3 setvar 𝑐
3		ccat 17601	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1539	. . . . 5 class 𝑐
6		cbs 17155	. . . . 5 class Base
7	5, 6	cfv 6499	. . . 4 class (Base‘𝑐)
8		vh	. . . . 5 setvar ℎ
9		chom 17207	. . . . . 6 class Hom
10	5, 9	cfv 6499	. . . . 5 class (Hom ‘𝑐)
11		vo	. . . . . 6 setvar 𝑜
12		cco 17208	. . . . . . 7 class comp
13	5, 12	cfv 6499	. . . . . 6 class (comp‘𝑐)
14		vx	. . . . . . 7 setvar 𝑥
15	4	cv 1539	. . . . . . 7 class 𝑏
16		vg	. . . . . . . . . . . . . 14 setvar 𝑔
17	16	cv 1539	. . . . . . . . . . . . 13 class 𝑔
18		vf	. . . . . . . . . . . . . 14 setvar 𝑓
19	18	cv 1539	. . . . . . . . . . . . 13 class 𝑓
20		vy	. . . . . . . . . . . . . . . 16 setvar 𝑦
21	20	cv 1539	. . . . . . . . . . . . . . 15 class 𝑦
22	14	cv 1539	. . . . . . . . . . . . . . 15 class 𝑥
23	21, 22	cop 4591	. . . . . . . . . . . . . 14 class ⟨𝑦, 𝑥⟩
24	11	cv 1539	. . . . . . . . . . . . . 14 class 𝑜
25	23, 22, 24	co 7369	. . . . . . . . . . . . 13 class (⟨𝑦, 𝑥⟩𝑜𝑥)
26	17, 19, 25	co 7369	. . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓)
27	26, 19	wceq 1540	. . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
28	8	cv 1539	. . . . . . . . . . . 12 class ℎ
29	21, 22, 28	co 7369	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
30	27, 18, 29	wral 3044	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
31	22, 22	cop 4591	. . . . . . . . . . . . . 14 class ⟨𝑥, 𝑥⟩
32	31, 21, 24	co 7369	. . . . . . . . . . . . 13 class (⟨𝑥, 𝑥⟩𝑜𝑦)
33	19, 17, 32	co 7369	. . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔)
34	33, 19	wceq 1540	. . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
35	22, 21, 28	co 7369	. . . . . . . . . . 11 class (𝑥ℎ𝑦)
36	34, 18, 35	wral 3044	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
37	30, 36	wa 395	. . . . . . . . 9 wff (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
38	37, 20, 15	wral 3044	. . . . . . . 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 7369	. . . . . . . 8 class (𝑥ℎ𝑥)
40	38, 16, 39	crio 7325	. . . . . . 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 5183	. . . . . 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 3859	. . . . 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 3859	. . . 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 3859	. . 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 5183	. 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 1540	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  17613  cidffn  17615