Definition df-cat 16329

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 16330. (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) / 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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}

Distinct variable group:   𝑏,𝑐,𝑓,𝑔,ℎ,𝑘,𝑜,𝑤,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-cat

Step	Hyp	Ref	Expression
1		ccat 16325	. 2 class Cat
2		vg	. . . . . . . . . . . . . . 15 setvar 𝑔
3	2	cv 1482	. . . . . . . . . . . . . 14 class 𝑔
4		vf	. . . . . . . . . . . . . . 15 setvar 𝑓
5	4	cv 1482	. . . . . . . . . . . . . 14 class 𝑓
6		vy	. . . . . . . . . . . . . . . . 17 setvar 𝑦
7	6	cv 1482	. . . . . . . . . . . . . . . 16 class 𝑦
8		vx	. . . . . . . . . . . . . . . . 17 setvar 𝑥
9	8	cv 1482	. . . . . . . . . . . . . . . 16 class 𝑥
10	7, 9	cop 4183	. . . . . . . . . . . . . . 15 class ⟨𝑦, 𝑥⟩
11		vo	. . . . . . . . . . . . . . . 16 setvar 𝑜
12	11	cv 1482	. . . . . . . . . . . . . . 15 class 𝑜
13	10, 9, 12	co 6650	. . . . . . . . . . . . . 14 class (⟨𝑦, 𝑥⟩𝑜𝑥)
14	3, 5, 13	co 6650	. . . . . . . . . . . . 13 class (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓)
15	14, 5	wceq 1483	. . . . . . . . . . . 12 wff (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
16		vh	. . . . . . . . . . . . . 14 setvar ℎ
17	16	cv 1482	. . . . . . . . . . . . 13 class ℎ
18	7, 9, 17	co 6650	. . . . . . . . . . . 12 class (𝑦ℎ𝑥)
19	15, 4, 18	wral 2912	. . . . . . . . . . 11 wff ∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓
20	9, 9	cop 4183	. . . . . . . . . . . . . . 15 class ⟨𝑥, 𝑥⟩
21	20, 7, 12	co 6650	. . . . . . . . . . . . . 14 class (⟨𝑥, 𝑥⟩𝑜𝑦)
22	5, 3, 21	co 6650	. . . . . . . . . . . . 13 class (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔)
23	22, 5	wceq 1483	. . . . . . . . . . . 12 wff (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
24	9, 7, 17	co 6650	. . . . . . . . . . . 12 class (𝑥ℎ𝑦)
25	23, 4, 24	wral 2912	. . . . . . . . . . 11 wff ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓
26	19, 25	wa 384	. . . . . . . . . 10 wff (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
27		vb	. . . . . . . . . . 11 setvar 𝑏
28	27	cv 1482	. . . . . . . . . 10 class 𝑏
29	26, 6, 28	wral 2912	. . . . . . . . 9 wff ∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
30	9, 9, 17	co 6650	. . . . . . . . 9 class (𝑥ℎ𝑥)
31	29, 2, 30	wrex 2913	. . . . . . . 8 wff ∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓)
32	9, 7	cop 4183	. . . . . . . . . . . . . . . 16 class ⟨𝑥, 𝑦⟩
33		vz	. . . . . . . . . . . . . . . . 17 setvar 𝑧
34	33	cv 1482	. . . . . . . . . . . . . . . 16 class 𝑧
35	32, 34, 12	co 6650	. . . . . . . . . . . . . . 15 class (⟨𝑥, 𝑦⟩𝑜𝑧)
36	3, 5, 35	co 6650	. . . . . . . . . . . . . 14 class (𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)
37	9, 34, 17	co 6650	. . . . . . . . . . . . . 14 class (𝑥ℎ𝑧)
38	36, 37	wcel 1990	. . . . . . . . . . . . 13 wff (𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧)
39		vk	. . . . . . . . . . . . . . . . . . 19 setvar 𝑘
40	39	cv 1482	. . . . . . . . . . . . . . . . . 18 class 𝑘
41	7, 34	cop 4183	. . . . . . . . . . . . . . . . . . 19 class ⟨𝑦, 𝑧⟩
42		vw	. . . . . . . . . . . . . . . . . . . 20 setvar 𝑤
43	42	cv 1482	. . . . . . . . . . . . . . . . . . 19 class 𝑤
44	41, 43, 12	co 6650	. . . . . . . . . . . . . . . . . 18 class (⟨𝑦, 𝑧⟩𝑜𝑤)
45	40, 3, 44	co 6650	. . . . . . . . . . . . . . . . 17 class (𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)
46	32, 43, 12	co 6650	. . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑦⟩𝑜𝑤)
47	45, 5, 46	co 6650	. . . . . . . . . . . . . . . 16 class ((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓)
48	9, 34	cop 4183	. . . . . . . . . . . . . . . . . 18 class ⟨𝑥, 𝑧⟩
49	48, 43, 12	co 6650	. . . . . . . . . . . . . . . . 17 class (⟨𝑥, 𝑧⟩𝑜𝑤)
50	40, 36, 49	co 6650	. . . . . . . . . . . . . . . 16 class (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))
51	47, 50	wceq 1483	. . . . . . . . . . . . . . 15 wff ((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))
52	34, 43, 17	co 6650	. . . . . . . . . . . . . . 15 class (𝑧ℎ𝑤)
53	51, 39, 52	wral 2912	. . . . . . . . . . . . . 14 wff ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))
54	53, 42, 28	wral 2912	. . . . . . . . . . . . 13 wff ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))
55	38, 54	wa 384	. . . . . . . . . . . 12 wff ((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))
56	7, 34, 17	co 6650	. . . . . . . . . . . 12 class (𝑦ℎ𝑧)
57	55, 2, 56	wral 2912	. . . . . . . . . . 11 wff ∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))
58	57, 4, 24	wral 2912	. . . . . . . . . 10 wff ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))
59	58, 33, 28	wral 2912	. . . . . . . . 9 wff ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))
60	59, 6, 28	wral 2912	. . . . . . . 8 wff ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))
61	31, 60	wa 384	. . . . . . 7 wff (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))
62	61, 8, 28	wral 2912	. . . . . 6 wff ∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))
63		vc	. . . . . . . 8 setvar 𝑐
64	63	cv 1482	. . . . . . 7 class 𝑐
65		cco 15953	. . . . . . 7 class comp
66	64, 65	cfv 5888	. . . . . 6 class (comp‘𝑐)
67	62, 11, 66	wsbc 3435	. . . . 5 wff [(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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))
68		chom 15952	. . . . . 6 class Hom
69	64, 68	cfv 5888	. . . . 5 class (Hom ‘𝑐)
70	67, 16, 69	wsbc 3435	. . . 4 wff [(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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))
71		cbs 15857	. . . . 5 class Base
72	64, 71	cfv 5888	. . . 4 class (Base‘𝑐)
73	70, 27, 72	wsbc 3435	. . 3 wff [(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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))
74	73, 63	cab 2608	. 2 class {𝑐 ∣ [(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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}
75	1, 74	wceq 1483	1 wff Cat = {𝑐 ∣ [(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) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}

This definition is referenced by:  iscat  16333