df-fuco - Mathbox for Zhi Wang

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Definition df-fuco 50369

Description: Definition of functor composition bifunctors. Given three categories 𝐶, 𝐷, and 𝐸, (⟨𝐶, 𝐷⟩ ∘_F 𝐸) is a functor from the product category of two categories of functors to a category of functors (fucofunc 50411). The object part maps two functors to their composition (fuco11 50378 and fuco11b 50389). The morphism part defines the "composition" of two natural transformations (fuco22 50391) into another natural transformation (fuco22nat 50398) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 50404). Note that such "composition" is different from fucco 18120 because they "compose" along different "axes". (Contributed by Zhi Wang, 29-Sep-2025.)

**Assertion**
Ref	Expression
df-fuco	⊢ ∘_F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Distinct variable group: 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑘,𝑙,𝑚,𝑝,𝑟,𝑢,𝑣,𝑤,𝑥

**Detailed syntax breakdown of Definition df-fuco**
Step	Hyp	Ref	Expression
1		cfuco 50368	. 2 class ∘_F
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3451	. . 3 class V
5		vc	. . . 4 setvar 𝑐
6	2	cv 1569	. . . . 5 class 𝑝
7		c1st 7988	. . . . 5 class 1^st
8	6, 7	cfv 6531	. . . 4 class (1^st ‘𝑝)
9		vd	. . . . 5 setvar 𝑑
10		c2nd 7989	. . . . . 6 class 2^nd
11	6, 10	cfv 6531	. . . . 5 class (2^nd ‘𝑝)
12		vw	. . . . . 6 setvar 𝑤
13	9	cv 1569	. . . . . . . 8 class 𝑑
14	3	cv 1569	. . . . . . . 8 class 𝑒
15		cfunc 18009	. . . . . . . 8 class Func
16	13, 14, 15	co 7412	. . . . . . 7 class (𝑑 Func 𝑒)
17	5	cv 1569	. . . . . . . 8 class 𝑐
18	17, 13, 15	co 7412	. . . . . . 7 class (𝑐 Func 𝑑)
19	16, 18	cxp 5649	. . . . . 6 class ((𝑑 Func 𝑒) × (𝑐 Func 𝑑))
20		ccofu 18011	. . . . . . . 8 class ∘_func
21	12	cv 1569	. . . . . . . 8 class 𝑤
22	20, 21	cres 5653	. . . . . . 7 class ( ∘_func ↾ 𝑤)
23		vu	. . . . . . . 8 setvar 𝑢
24		vv	. . . . . . . 8 setvar 𝑣
25		vf	. . . . . . . . 9 setvar 𝑓
26	23	cv 1569	. . . . . . . . . . 11 class 𝑢
27	26, 10	cfv 6531	. . . . . . . . . 10 class (2^nd ‘𝑢)
28	27, 7	cfv 6531	. . . . . . . . 9 class (1^st ‘(2^nd ‘𝑢))
29		vk	. . . . . . . . . 10 setvar 𝑘
30	26, 7	cfv 6531	. . . . . . . . . . 11 class (1^st ‘𝑢)
31	30, 7	cfv 6531	. . . . . . . . . 10 class (1^st ‘(1^st ‘𝑢))
32		vl	. . . . . . . . . . 11 setvar 𝑙
33	30, 10	cfv 6531	. . . . . . . . . . 11 class (2^nd ‘(1^st ‘𝑢))
34		vm	. . . . . . . . . . . 12 setvar 𝑚
35	24	cv 1569	. . . . . . . . . . . . . 14 class 𝑣
36	35, 10	cfv 6531	. . . . . . . . . . . . 13 class (2^nd ‘𝑣)
37	36, 7	cfv 6531	. . . . . . . . . . . 12 class (1^st ‘(2^nd ‘𝑣))
38		vr	. . . . . . . . . . . . 13 setvar 𝑟
39	35, 7	cfv 6531	. . . . . . . . . . . . . 14 class (1^st ‘𝑣)
40	39, 7	cfv 6531	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑣))
41		vb	. . . . . . . . . . . . . 14 setvar 𝑏
42		va	. . . . . . . . . . . . . 14 setvar 𝑎
43		cnat 18099	. . . . . . . . . . . . . . . 16 class Nat
44	13, 14, 43	co 7412	. . . . . . . . . . . . . . 15 class (𝑑 Nat 𝑒)
45	30, 39, 44	co 7412	. . . . . . . . . . . . . 14 class ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣))
46	17, 13, 43	co 7412	. . . . . . . . . . . . . . 15 class (𝑐 Nat 𝑑)
47	27, 36, 46	co 7412	. . . . . . . . . . . . . 14 class ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣))
48		vx	. . . . . . . . . . . . . . 15 setvar 𝑥
49		cbs 17367	. . . . . . . . . . . . . . . 16 class Base
50	17, 49	cfv 6531	. . . . . . . . . . . . . . 15 class (Base‘𝑐)
51	48	cv 1569	. . . . . . . . . . . . . . . . . 18 class 𝑥
52	34	cv 1569	. . . . . . . . . . . . . . . . . 18 class 𝑚
53	51, 52	cfv 6531	. . . . . . . . . . . . . . . . 17 class (𝑚‘𝑥)
54	41	cv 1569	. . . . . . . . . . . . . . . . 17 class 𝑏
55	53, 54	cfv 6531	. . . . . . . . . . . . . . . 16 class (𝑏‘(𝑚‘𝑥))
56	42	cv 1569	. . . . . . . . . . . . . . . . . 18 class 𝑎
57	51, 56	cfv 6531	. . . . . . . . . . . . . . . . 17 class (𝑎‘𝑥)
58	25	cv 1569	. . . . . . . . . . . . . . . . . . 19 class 𝑓
59	51, 58	cfv 6531	. . . . . . . . . . . . . . . . . 18 class (𝑓‘𝑥)
60	32	cv 1569	. . . . . . . . . . . . . . . . . 18 class 𝑙
61	59, 53, 60	co 7412	. . . . . . . . . . . . . . . . 17 class ((𝑓‘𝑥)𝑙(𝑚‘𝑥))
62	57, 61	cfv 6531	. . . . . . . . . . . . . . . 16 class (((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))
63	29	cv 1569	. . . . . . . . . . . . . . . . . . 19 class 𝑘
64	59, 63	cfv 6531	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑓‘𝑥))
65	53, 63	cfv 6531	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑚‘𝑥))
66	64, 65	cop 4590	. . . . . . . . . . . . . . . . 17 class ⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩
67	38	cv 1569	. . . . . . . . . . . . . . . . . 18 class 𝑟
68	53, 67	cfv 6531	. . . . . . . . . . . . . . . . 17 class (𝑟‘(𝑚‘𝑥))
69		cco 17420	. . . . . . . . . . . . . . . . . 18 class comp
70	14, 69	cfv 6531	. . . . . . . . . . . . . . . . 17 class (comp‘𝑒)
71	66, 68, 70	co 7412	. . . . . . . . . . . . . . . 16 class (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))
72	55, 62, 71	co 7412	. . . . . . . . . . . . . . 15 class ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))
73	48, 50, 72	cmpt 5186	. . . . . . . . . . . . . 14 class (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))
74	41, 42, 45, 47, 73	cmpo 7414	. . . . . . . . . . . . 13 class (𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
75	38, 40, 74	csb 3847	. . . . . . . . . . . 12 class ⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
76	34, 37, 75	csb 3847	. . . . . . . . . . 11 class ⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
77	32, 33, 76	csb 3847	. . . . . . . . . 10 class ⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
78	29, 31, 77	csb 3847	. . . . . . . . 9 class ⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
79	25, 28, 78	csb 3847	. . . . . . . 8 class ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
80	23, 24, 21, 21, 79	cmpo 7414	. . . . . . 7 class (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))
81	22, 80	cop 4590	. . . . . 6 class ⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩
82	12, 19, 81	csb 3847	. . . . 5 class ⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩
83	9, 11, 82	csb 3847	. . . 4 class ⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩
84	5, 8, 83	csb 3847	. . 3 class ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩
85	2, 3, 4, 4, 84	cmpo 7414	. 2 class (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
86	1, 85	wceq 1570	1 wff ∘_F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Colors of variables: wff setvar class

This definition is used by: fucofvalg 50370

W3C validator