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Definition df-cid 16377
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 16373 . 2 class Id
2 vc . . 3 setvar 𝑐
3 ccat 16372 . . 3 class Cat
4 vb . . . 4 setvar 𝑏
52cv 1522 . . . . 5 class 𝑐
6 cbs 15904 . . . . 5 class Base
75, 6cfv 5926 . . . 4 class (Base‘𝑐)
8 vh . . . . 5 setvar
9 chom 15999 . . . . . 6 class Hom
105, 9cfv 5926 . . . . 5 class (Hom ‘𝑐)
11 vo . . . . . 6 setvar 𝑜
12 cco 16000 . . . . . . 7 class comp
135, 12cfv 5926 . . . . . 6 class (comp‘𝑐)
14 vx . . . . . . 7 setvar 𝑥
154cv 1522 . . . . . . 7 class 𝑏
16 vg . . . . . . . . . . . . . 14 setvar 𝑔
1716cv 1522 . . . . . . . . . . . . 13 class 𝑔
18 vf . . . . . . . . . . . . . 14 setvar 𝑓
1918cv 1522 . . . . . . . . . . . . 13 class 𝑓
20 vy . . . . . . . . . . . . . . . 16 setvar 𝑦
2120cv 1522 . . . . . . . . . . . . . . 15 class 𝑦
2214cv 1522 . . . . . . . . . . . . . . 15 class 𝑥
2321, 22cop 4216 . . . . . . . . . . . . . 14 class 𝑦, 𝑥
2411cv 1522 . . . . . . . . . . . . . 14 class 𝑜
2523, 22, 24co 6690 . . . . . . . . . . . . 13 class (⟨𝑦, 𝑥𝑜𝑥)
2617, 19, 25co 6690 . . . . . . . . . . . 12 class (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓)
2726, 19wceq 1523 . . . . . . . . . . 11 wff (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
288cv 1522 . . . . . . . . . . . 12 class
2921, 22, 28co 6690 . . . . . . . . . . 11 class (𝑦𝑥)
3027, 18, 29wral 2941 . . . . . . . . . 10 wff 𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
3122, 22cop 4216 . . . . . . . . . . . . . 14 class 𝑥, 𝑥
3231, 21, 24co 6690 . . . . . . . . . . . . 13 class (⟨𝑥, 𝑥𝑜𝑦)
3319, 17, 32co 6690 . . . . . . . . . . . 12 class (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔)
3433, 19wceq 1523 . . . . . . . . . . 11 wff (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
3522, 21, 28co 6690 . . . . . . . . . . 11 class (𝑥𝑦)
3634, 18, 35wral 2941 . . . . . . . . . 10 wff 𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
3730, 36wa 383 . . . . . . . . 9 wff (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
3837, 20, 15wral 2941 . . . . . . . 8 wff 𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
3922, 22, 28co 6690 . . . . . . . 8 class (𝑥𝑥)
4038, 16, 39crio 6650 . . . . . . 7 class (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))
4114, 15, 40cmpt 4762 . . . . . 6 class (𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)))
4211, 13, 41csb 3566 . . . . 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 3566 . . . 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 3566 . . 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 4762 . 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 1523 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  16384  cidffn  16386
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