Definition df-cid 17637

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 17633	. 2 class Id
2		vc	. . 3 setvar 𝑐
3		ccat 17632	. . 3 class Cat
4		vb	. . . 4 setvar 𝑏
5	2	cv 1539	. . . . 5 class 𝑐
6		cbs 17186	. . . . 5 class Base
7	5, 6	cfv 6514	. . . 4 class (Base‘𝑐)
8		vh	. . . . 5 setvar ℎ
9		chom 17238	. . . . . 6 class Hom
10	5, 9	cfv 6514	. . . . 5 class (Hom ‘𝑐)
11		vo	. . . . . 6 setvar 𝑜
12		cco 17239	. . . . . . 7 class comp
13	5, 12	cfv 6514	. . . . . 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 4598	. . . . . . . . . . . . . 14 class ⟨𝑦, 𝑥⟩
24	11	cv 1539	. . . . . . . . . . . . . 14 class 𝑜
25	23, 22, 24	co 7390	. . . . . . . . . . . . 13 class (⟨𝑦, 𝑥⟩𝑜𝑥)
26	17, 19, 25	co 7390	. . . . . . . . . . . 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 7390	. . . . . . . . . . 11 class (𝑦ℎ𝑥)
30	27, 18, 29	wral 3045	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
31	22, 22	cop 4598	. . . . . . . . . . . . . 14 class ⟨𝑥, 𝑥⟩
32	31, 21, 24	co 7390	. . . . . . . . . . . . 13 class (⟨𝑥, 𝑥⟩𝑜𝑦)
33	19, 17, 32	co 7390	. . . . . . . . . . . 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 7390	. . . . . . . . . . 11 class (𝑥ℎ𝑦)
36	34, 18, 35	wral 3045	. . . . . . . . . 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 3045	. . . . . . . 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 7390	. . . . . . . 8 class (𝑥ℎ𝑥)
40	38, 16, 39	crio 7346	. . . . . . 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 5191	. . . . . 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 3865	. . . . 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 3865	. . . 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 3865	. . 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 5191	. 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  17644  cidffn  17646