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Definition df-cid 16931
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) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))
Distinct variable group:   𝑏,𝑐,𝑓,𝑔,,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-cid
StepHypRef Expression
1 ccid 16927 . 2 class Id
2 vc . . 3 setvar 𝑐
3 ccat 16926 . . 3 class Cat
4 vb . . . 4 setvar 𝑏
52cv 1537 . . . . 5 class 𝑐
6 cbs 16474 . . . . 5 class Base
75, 6cfv 6334 . . . 4 class (Base‘𝑐)
8 vh . . . . 5 setvar
9 chom 16567 . . . . . 6 class Hom
105, 9cfv 6334 . . . . 5 class (Hom ‘𝑐)
11 vo . . . . . 6 setvar 𝑜
12 cco 16568 . . . . . . 7 class comp
135, 12cfv 6334 . . . . . 6 class (comp‘𝑐)
14 vx . . . . . . 7 setvar 𝑥
154cv 1537 . . . . . . 7 class 𝑏
16 vg . . . . . . . . . . . . . 14 setvar 𝑔
1716cv 1537 . . . . . . . . . . . . 13 class 𝑔
18 vf . . . . . . . . . . . . . 14 setvar 𝑓
1918cv 1537 . . . . . . . . . . . . 13 class 𝑓
20 vy . . . . . . . . . . . . . . . 16 setvar 𝑦
2120cv 1537 . . . . . . . . . . . . . . 15 class 𝑦
2214cv 1537 . . . . . . . . . . . . . . 15 class 𝑥
2321, 22cop 4545 . . . . . . . . . . . . . 14 class 𝑦, 𝑥
2411cv 1537 . . . . . . . . . . . . . 14 class 𝑜
2523, 22, 24co 7140 . . . . . . . . . . . . 13 class (⟨𝑦, 𝑥𝑜𝑥)
2617, 19, 25co 7140 . . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓)
2726, 19wceq 1538 . . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
288cv 1537 . . . . . . . . . . . 12 class
2921, 22, 28co 7140 . . . . . . . . . . 11 class (𝑦𝑥)
3027, 18, 29wral 3130 . . . . . . . . . 10 wff 𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
3122, 22cop 4545 . . . . . . . . . . . . . 14 class 𝑥, 𝑥
3231, 21, 24co 7140 . . . . . . . . . . . . 13 class (⟨𝑥, 𝑥𝑜𝑦)
3319, 17, 32co 7140 . . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔)
3433, 19wceq 1538 . . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
3522, 21, 28co 7140 . . . . . . . . . . 11 class (𝑥𝑦)
3634, 18, 35wral 3130 . . . . . . . . . 10 wff 𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
3730, 36wa 399 . . . . . . . . 9 wff (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
3837, 20, 15wral 3130 . . . . . . . 8 wff 𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
3922, 22, 28co 7140 . . . . . . . 8 class (𝑥𝑥)
4038, 16, 39crio 7097 . . . . . . 7 class (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))
4114, 15, 40cmpt 5122 . . . . . 6 class (𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)))
4211, 13, 41csb 3855 . . . . 5 class (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)))
438, 10, 42csb 3855 . . . 4 class (Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)))
444, 7, 43csb 3855 . . 3 class (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)))
452, 3, 44cmpt 5122 . 2 class (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))
461, 45wceq 1538 1 wff Id = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))
Colors of variables: wff setvar class
This definition is referenced by:  cidfval  16938  cidffn  16940
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