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Definition df-cat 16550
Description: A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 16551. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
df-cat Cat = {𝑐[(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) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}
Distinct variable group:   𝑏,𝑐,𝑓,𝑔,,𝑘,𝑜,𝑤,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-cat
StepHypRef Expression
1 ccat 16546 . 2 class Cat
2 vg . . . . . . . . . . . . . . 15 setvar 𝑔
32cv 1631 . . . . . . . . . . . . . 14 class 𝑔
4 vf . . . . . . . . . . . . . . 15 setvar 𝑓
54cv 1631 . . . . . . . . . . . . . 14 class 𝑓
6 vy . . . . . . . . . . . . . . . . 17 setvar 𝑦
76cv 1631 . . . . . . . . . . . . . . . 16 class 𝑦
8 vx . . . . . . . . . . . . . . . . 17 setvar 𝑥
98cv 1631 . . . . . . . . . . . . . . . 16 class 𝑥
107, 9cop 4327 . . . . . . . . . . . . . . 15 class 𝑦, 𝑥
11 vo . . . . . . . . . . . . . . . 16 setvar 𝑜
1211cv 1631 . . . . . . . . . . . . . . 15 class 𝑜
1310, 9, 12co 6814 . . . . . . . . . . . . . 14 class (⟨𝑦, 𝑥𝑜𝑥)
143, 5, 13co 6814 . . . . . . . . . . . . 13 class (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓)
1514, 5wceq 1632 . . . . . . . . . . . 12 wff (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
16 vh . . . . . . . . . . . . . 14 setvar
1716cv 1631 . . . . . . . . . . . . 13 class
187, 9, 17co 6814 . . . . . . . . . . . 12 class (𝑦𝑥)
1915, 4, 18wral 3050 . . . . . . . . . . 11 wff 𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
209, 9cop 4327 . . . . . . . . . . . . . . 15 class 𝑥, 𝑥
2120, 7, 12co 6814 . . . . . . . . . . . . . 14 class (⟨𝑥, 𝑥𝑜𝑦)
225, 3, 21co 6814 . . . . . . . . . . . . 13 class (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔)
2322, 5wceq 1632 . . . . . . . . . . . 12 wff (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
249, 7, 17co 6814 . . . . . . . . . . . 12 class (𝑥𝑦)
2523, 4, 24wral 3050 . . . . . . . . . . 11 wff 𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
2619, 25wa 383 . . . . . . . . . 10 wff (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
27 vb . . . . . . . . . . 11 setvar 𝑏
2827cv 1631 . . . . . . . . . 10 class 𝑏
2926, 6, 28wral 3050 . . . . . . . . 9 wff 𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
309, 9, 17co 6814 . . . . . . . . 9 class (𝑥𝑥)
3129, 2, 30wrex 3051 . . . . . . . 8 wff 𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
329, 7cop 4327 . . . . . . . . . . . . . . . 16 class 𝑥, 𝑦
33 vz . . . . . . . . . . . . . . . . 17 setvar 𝑧
3433cv 1631 . . . . . . . . . . . . . . . 16 class 𝑧
3532, 34, 12co 6814 . . . . . . . . . . . . . . 15 class (⟨𝑥, 𝑦𝑜𝑧)
363, 5, 35co 6814 . . . . . . . . . . . . . 14 class (𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)
379, 34, 17co 6814 . . . . . . . . . . . . . 14 class (𝑥𝑧)
3836, 37wcel 2139 . . . . . . . . . . . . 13 wff (𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧)
39 vk . . . . . . . . . . . . . . . . . . 19 setvar 𝑘
4039cv 1631 . . . . . . . . . . . . . . . . . 18 class 𝑘
417, 34cop 4327 . . . . . . . . . . . . . . . . . . 19 class 𝑦, 𝑧
42 vw . . . . . . . . . . . . . . . . . . . 20 setvar 𝑤
4342cv 1631 . . . . . . . . . . . . . . . . . . 19 class 𝑤
4441, 43, 12co 6814 . . . . . . . . . . . . . . . . . 18 class (⟨𝑦, 𝑧𝑜𝑤)
4540, 3, 44co 6814 . . . . . . . . . . . . . . . . 17 class (𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)
4632, 43, 12co 6814 . . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑦𝑜𝑤)
4745, 5, 46co 6814 . . . . . . . . . . . . . . . 16 class ((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓)
489, 34cop 4327 . . . . . . . . . . . . . . . . . 18 class 𝑥, 𝑧
4948, 43, 12co 6814 . . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑧𝑜𝑤)
5040, 36, 49co 6814 . . . . . . . . . . . . . . . 16 class (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5147, 50wceq 1632 . . . . . . . . . . . . . . 15 wff ((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5234, 43, 17co 6814 . . . . . . . . . . . . . . 15 class (𝑧𝑤)
5351, 39, 52wral 3050 . . . . . . . . . . . . . 14 wff 𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5453, 42, 28wral 3050 . . . . . . . . . . . . 13 wff 𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5538, 54wa 383 . . . . . . . . . . . 12 wff ((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)))
567, 34, 17co 6814 . . . . . . . . . . . 12 class (𝑦𝑧)
5755, 2, 56wral 3050 . . . . . . . . . . 11 wff 𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)))
5857, 4, 24wral 3050 . . . . . . . . . 10 wff 𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)))
5958, 33, 28wral 3050 . . . . . . . . 9 wff 𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)))
6059, 6, 28wral 3050 . . . . . . . 8 wff 𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)))
6131, 60wa 383 . . . . . . 7 wff (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))
6261, 8, 28wral 3050 . . . . . 6 wff 𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))
63 vc . . . . . . . 8 setvar 𝑐
6463cv 1631 . . . . . . 7 class 𝑐
65 cco 16175 . . . . . . 7 class comp
6664, 65cfv 6049 . . . . . 6 class (comp‘𝑐)
6762, 11, 66wsbc 3576 . . . . 5 wff [(comp‘𝑐) / 𝑜]𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))
68 chom 16174 . . . . . 6 class Hom
6964, 68cfv 6049 . . . . 5 class (Hom ‘𝑐)
7067, 16, 69wsbc 3576 . . . 4 wff [(Hom ‘𝑐) / ][(comp‘𝑐) / 𝑜]𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))
71 cbs 16079 . . . . 5 class Base
7264, 71cfv 6049 . . . 4 class (Base‘𝑐)
7370, 27, 72wsbc 3576 . . 3 wff [(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) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))
7473, 63cab 2746 . 2 class {𝑐[(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) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}
751, 74wceq 1632 1 wff Cat = {𝑐[(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) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}
Colors of variables: wff setvar class
This definition is referenced by:  iscat  16554
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