Definition df-cid 17726

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 17722	. 2 class Id
2		vc	. . 3 setvar 𝑐
3		ccat 17721	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1569	. . . . 5 class 𝑐
6		cbs 17270	. . . . 5 class Base
7	5, 6	cfv 6538	. . . 4 class (Base‘𝑐)
8		vh	. . . . 5 setvar ℎ
9		chom 17322	. . . . . 6 class Hom
10	5, 9	cfv 6538	. . . . 5 class (Hom ‘𝑐)
11		vo	. . . . . 6 setvar 𝑜
12		cco 17323	. . . . . . 7 class comp
13	5, 12	cfv 6538	. . . . . 6 class (comp‘𝑐)
14		vx	. . . . . . 7 setvar 𝑥
15	4	cv 1569	. . . . . . 7 class 𝑏
16		vg	. . . . . . . . . . . . . 14 setvar 𝑔
17	16	cv 1569	. . . . . . . . . . . . 13 class 𝑔
18		vf	. . . . . . . . . . . . . 14 setvar 𝑓
19	18	cv 1569	. . . . . . . . . . . . 13 class 𝑓
20		vy	. . . . . . . . . . . . . . . 16 setvar 𝑦
21	20	cv 1569	. . . . . . . . . . . . . . 15 class 𝑦
22	14	cv 1569	. . . . . . . . . . . . . . 15 class 𝑥
23	21, 22	cop 4596	. . . . . . . . . . . . . 14 class ⟨𝑦, 𝑥⟩
24	11	cv 1569	. . . . . . . . . . . . . 14 class 𝑜
25	23, 22, 24	co 7412	. . . . . . . . . . . . 13 class (⟨𝑦, 𝑥⟩𝑜𝑥)
26	17, 19, 25	co 7412	. . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓)
27	26, 19	wceq 1570	. . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
28	8	cv 1569	. . . . . . . . . . . 12 class ℎ
29	21, 22, 28	co 7412	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
30	27, 18, 29	wral 3079	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
31	22, 22	cop 4596	. . . . . . . . . . . . . 14 class ⟨𝑥, 𝑥⟩
32	31, 21, 24	co 7412	. . . . . . . . . . . . 13 class (⟨𝑥, 𝑥⟩𝑜𝑦)
33	19, 17, 32	co 7412	. . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔)
34	33, 19	wceq 1570	. . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
35	22, 21, 28	co 7412	. . . . . . . . . . 11 class (𝑥ℎ𝑦)
36	34, 18, 35	wral 3079	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
37	30, 36	wa 400	. . . . . . . . 9 wff (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
38	37, 20, 15	wral 3079	. . . . . . . 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 7412	. . . . . . . 8 class (𝑥ℎ𝑥)
40	38, 16, 39	crio 7368	. . . . . . 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 3854	. . . . 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 3854	. . . 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 3854	. . 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 1570	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  17733  cidffn  17735