Definition df-cid 16940

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 16936	. 2 class Id
2		vc	. . 3 setvar 𝑐
3		ccat 16935	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1536	. . . . 5 class 𝑐
6		cbs 16483	. . . . 5 class Base
7	5, 6	cfv 6355	. . . 4 class (Base‘𝑐)
8		vh	. . . . 5 setvar ℎ
9		chom 16576	. . . . . 6 class Hom
10	5, 9	cfv 6355	. . . . 5 class (Hom ‘𝑐)
11		vo	. . . . . 6 setvar 𝑜
12		cco 16577	. . . . . . 7 class comp
13	5, 12	cfv 6355	. . . . . 6 class (comp‘𝑐)
14		vx	. . . . . . 7 setvar 𝑥
15	4	cv 1536	. . . . . . 7 class 𝑏
16		vg	. . . . . . . . . . . . . 14 setvar 𝑔
17	16	cv 1536	. . . . . . . . . . . . 13 class 𝑔
18		vf	. . . . . . . . . . . . . 14 setvar 𝑓
19	18	cv 1536	. . . . . . . . . . . . 13 class 𝑓
20		vy	. . . . . . . . . . . . . . . 16 setvar 𝑦
21	20	cv 1536	. . . . . . . . . . . . . . 15 class 𝑦
22	14	cv 1536	. . . . . . . . . . . . . . 15 class 𝑥
23	21, 22	cop 4573	. . . . . . . . . . . . . 14 class ⟨𝑦, 𝑥⟩
24	11	cv 1536	. . . . . . . . . . . . . 14 class 𝑜
25	23, 22, 24	co 7156	. . . . . . . . . . . . 13 class (⟨𝑦, 𝑥⟩𝑜𝑥)
26	17, 19, 25	co 7156	. . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓)
27	26, 19	wceq 1537	. . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
28	8	cv 1536	. . . . . . . . . . . 12 class ℎ
29	21, 22, 28	co 7156	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
30	27, 18, 29	wral 3138	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
31	22, 22	cop 4573	. . . . . . . . . . . . . 14 class ⟨𝑥, 𝑥⟩
32	31, 21, 24	co 7156	. . . . . . . . . . . . 13 class (⟨𝑥, 𝑥⟩𝑜𝑦)
33	19, 17, 32	co 7156	. . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔)
34	33, 19	wceq 1537	. . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
35	22, 21, 28	co 7156	. . . . . . . . . . 11 class (𝑥ℎ𝑦)
36	34, 18, 35	wral 3138	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
37	30, 36	wa 398	. . . . . . . . 9 wff (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
38	37, 20, 15	wral 3138	. . . . . . . 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 7156	. . . . . . . 8 class (𝑥ℎ𝑥)
40	38, 16, 39	crio 7113	. . . . . . 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 5146	. . . . . 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 3883	. . . . 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 3883	. . . 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 3883	. . 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 5146	. 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 1537	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  16947  cidffn  16949