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Definition df-cat 17711
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, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 18142). See setc2obas 18139 and setc2ohom 18140 for a counterexample. 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 17712. (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 17707 . 2 class Cat
2 vg . . . . . . . . . . . . . . 15 setvar 𝑔
32cv 1539 . . . . . . . . . . . . . 14 class 𝑔
4 vf . . . . . . . . . . . . . . 15 setvar 𝑓
54cv 1539 . . . . . . . . . . . . . 14 class 𝑓
6 vy . . . . . . . . . . . . . . . . 17 setvar 𝑦
76cv 1539 . . . . . . . . . . . . . . . 16 class 𝑦
8 vx . . . . . . . . . . . . . . . . 17 setvar 𝑥
98cv 1539 . . . . . . . . . . . . . . . 16 class 𝑥
107, 9cop 4632 . . . . . . . . . . . . . . 15 class 𝑦, 𝑥
11 vo . . . . . . . . . . . . . . . 16 setvar 𝑜
1211cv 1539 . . . . . . . . . . . . . . 15 class 𝑜
1310, 9, 12co 7431 . . . . . . . . . . . . . 14 class (⟨𝑦, 𝑥𝑜𝑥)
143, 5, 13co 7431 . . . . . . . . . . . . 13 class (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓)
1514, 5wceq 1540 . . . . . . . . . . . 12 wff (𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
16 vh . . . . . . . . . . . . . 14 setvar
1716cv 1539 . . . . . . . . . . . . 13 class
187, 9, 17co 7431 . . . . . . . . . . . 12 class (𝑦𝑥)
1915, 4, 18wral 3061 . . . . . . . . . . 11 wff 𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓
209, 9cop 4632 . . . . . . . . . . . . . . 15 class 𝑥, 𝑥
2120, 7, 12co 7431 . . . . . . . . . . . . . 14 class (⟨𝑥, 𝑥𝑜𝑦)
225, 3, 21co 7431 . . . . . . . . . . . . 13 class (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔)
2322, 5wceq 1540 . . . . . . . . . . . 12 wff (𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
249, 7, 17co 7431 . . . . . . . . . . . 12 class (𝑥𝑦)
2523, 4, 24wral 3061 . . . . . . . . . . 11 wff 𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓
2619, 25wa 395 . . . . . . . . . 10 wff (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
27 vb . . . . . . . . . . 11 setvar 𝑏
2827cv 1539 . . . . . . . . . 10 class 𝑏
2926, 6, 28wral 3061 . . . . . . . . 9 wff 𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
309, 9, 17co 7431 . . . . . . . . 9 class (𝑥𝑥)
3129, 2, 30wrex 3070 . . . . . . . 8 wff 𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓)
329, 7cop 4632 . . . . . . . . . . . . . . . 16 class 𝑥, 𝑦
33 vz . . . . . . . . . . . . . . . . 17 setvar 𝑧
3433cv 1539 . . . . . . . . . . . . . . . 16 class 𝑧
3532, 34, 12co 7431 . . . . . . . . . . . . . . 15 class (⟨𝑥, 𝑦𝑜𝑧)
363, 5, 35co 7431 . . . . . . . . . . . . . 14 class (𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓)
379, 34, 17co 7431 . . . . . . . . . . . . . 14 class (𝑥𝑧)
3836, 37wcel 2108 . . . . . . . . . . . . 13 wff (𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧)
39 vk . . . . . . . . . . . . . . . . . . 19 setvar 𝑘
4039cv 1539 . . . . . . . . . . . . . . . . . 18 class 𝑘
417, 34cop 4632 . . . . . . . . . . . . . . . . . . 19 class 𝑦, 𝑧
42 vw . . . . . . . . . . . . . . . . . . . 20 setvar 𝑤
4342cv 1539 . . . . . . . . . . . . . . . . . . 19 class 𝑤
4441, 43, 12co 7431 . . . . . . . . . . . . . . . . . 18 class (⟨𝑦, 𝑧𝑜𝑤)
4540, 3, 44co 7431 . . . . . . . . . . . . . . . . 17 class (𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)
4632, 43, 12co 7431 . . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑦𝑜𝑤)
4745, 5, 46co 7431 . . . . . . . . . . . . . . . 16 class ((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓)
489, 34cop 4632 . . . . . . . . . . . . . . . . . 18 class 𝑥, 𝑧
4948, 43, 12co 7431 . . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑧𝑜𝑤)
5040, 36, 49co 7431 . . . . . . . . . . . . . . . 16 class (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5147, 50wceq 1540 . . . . . . . . . . . . . . 15 wff ((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5234, 43, 17co 7431 . . . . . . . . . . . . . . 15 class (𝑧𝑤)
5351, 39, 52wral 3061 . . . . . . . . . . . . . 14 wff 𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5453, 42, 28wral 3061 . . . . . . . . . . . . 13 wff 𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))
5538, 54wa 395 . . . . . . . . . . . 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 7431 . . . . . . . . . . . 12 class (𝑦𝑧)
5755, 2, 56wral 3061 . . . . . . . . . . 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 3061 . . . . . . . . . 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 3061 . . . . . . . . 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 3061 . . . . . . . 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 395 . . . . . . 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 3061 . . . . . 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 1539 . . . . . . 7 class 𝑐
65 cco 17309 . . . . . . 7 class comp
6664, 65cfv 6561 . . . . . 6 class (comp‘𝑐)
6762, 11, 66wsbc 3788 . . . . 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 17308 . . . . . 6 class Hom
6964, 68cfv 6561 . . . . 5 class (Hom ‘𝑐)
7067, 16, 69wsbc 3788 . . . 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 17247 . . . . 5 class Base
7264, 71cfv 6561 . . . 4 class (Base‘𝑐)
7370, 27, 72wsbc 3788 . . 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 2714 . 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 1540 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  17715
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