df-fuco - Mathbox for Zhi Wang

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Definition df-fuco 49348

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 49390). The object part maps two functors to their composition (fuco11 49357 and fuco11b 49368). The morphism part defines the "composition" of two natural transformations (fuco22 49370) into another natural transformation (fuco22nat 49377) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49383). Note that such "composition" is different from fucco 17869 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 49347	. 2 class ∘_F
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3436	. . 3 class V
5		vc	. . . 4 setvar 𝑐
6	2	cv 1540	. . . . 5 class 𝑝
7		c1st 7919	. . . . 5 class 1^st
8	6, 7	cfv 6481	. . . 4 class (1^st ‘𝑝)
9		vd	. . . . 5 setvar 𝑑
10		c2nd 7920	. . . . . 6 class 2^nd
11	6, 10	cfv 6481	. . . . 5 class (2^nd ‘𝑝)
12		vw	. . . . . 6 setvar 𝑤
13	9	cv 1540	. . . . . . . 8 class 𝑑
14	3	cv 1540	. . . . . . . 8 class 𝑒
15		cfunc 17758	. . . . . . . 8 class Func
16	13, 14, 15	co 7346	. . . . . . 7 class (𝑑 Func 𝑒)
17	5	cv 1540	. . . . . . . 8 class 𝑐
18	17, 13, 15	co 7346	. . . . . . 7 class (𝑐 Func 𝑑)
19	16, 18	cxp 5614	. . . . . 6 class ((𝑑 Func 𝑒) × (𝑐 Func 𝑑))
20		ccofu 17760	. . . . . . . 8 class ∘_func
21	12	cv 1540	. . . . . . . 8 class 𝑤
22	20, 21	cres 5618	. . . . . . 7 class ( ∘_func ↾ 𝑤)
23		vu	. . . . . . . 8 setvar 𝑢
24		vv	. . . . . . . 8 setvar 𝑣
25		vf	. . . . . . . . 9 setvar 𝑓
26	23	cv 1540	. . . . . . . . . . 11 class 𝑢
27	26, 10	cfv 6481	. . . . . . . . . 10 class (2^nd ‘𝑢)
28	27, 7	cfv 6481	. . . . . . . . 9 class (1^st ‘(2^nd ‘𝑢))
29		vk	. . . . . . . . . 10 setvar 𝑘
30	26, 7	cfv 6481	. . . . . . . . . . 11 class (1^st ‘𝑢)
31	30, 7	cfv 6481	. . . . . . . . . 10 class (1^st ‘(1^st ‘𝑢))
32		vl	. . . . . . . . . . 11 setvar 𝑙
33	30, 10	cfv 6481	. . . . . . . . . . 11 class (2^nd ‘(1^st ‘𝑢))
34		vm	. . . . . . . . . . . 12 setvar 𝑚
35	24	cv 1540	. . . . . . . . . . . . . 14 class 𝑣
36	35, 10	cfv 6481	. . . . . . . . . . . . 13 class (2^nd ‘𝑣)
37	36, 7	cfv 6481	. . . . . . . . . . . 12 class (1^st ‘(2^nd ‘𝑣))
38		vr	. . . . . . . . . . . . 13 setvar 𝑟
39	35, 7	cfv 6481	. . . . . . . . . . . . . 14 class (1^st ‘𝑣)
40	39, 7	cfv 6481	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑣))
41		vb	. . . . . . . . . . . . . 14 setvar 𝑏
42		va	. . . . . . . . . . . . . 14 setvar 𝑎
43		cnat 17848	. . . . . . . . . . . . . . . 16 class Nat
44	13, 14, 43	co 7346	. . . . . . . . . . . . . . 15 class (𝑑 Nat 𝑒)
45	30, 39, 44	co 7346	. . . . . . . . . . . . . 14 class ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣))
46	17, 13, 43	co 7346	. . . . . . . . . . . . . . 15 class (𝑐 Nat 𝑑)
47	27, 36, 46	co 7346	. . . . . . . . . . . . . 14 class ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣))
48		vx	. . . . . . . . . . . . . . 15 setvar 𝑥
49		cbs 17117	. . . . . . . . . . . . . . . 16 class Base
50	17, 49	cfv 6481	. . . . . . . . . . . . . . 15 class (Base‘𝑐)
51	48	cv 1540	. . . . . . . . . . . . . . . . . 18 class 𝑥
52	34	cv 1540	. . . . . . . . . . . . . . . . . 18 class 𝑚
53	51, 52	cfv 6481	. . . . . . . . . . . . . . . . 17 class (𝑚‘𝑥)
54	41	cv 1540	. . . . . . . . . . . . . . . . 17 class 𝑏
55	53, 54	cfv 6481	. . . . . . . . . . . . . . . 16 class (𝑏‘(𝑚‘𝑥))
56	42	cv 1540	. . . . . . . . . . . . . . . . . 18 class 𝑎
57	51, 56	cfv 6481	. . . . . . . . . . . . . . . . 17 class (𝑎‘𝑥)
58	25	cv 1540	. . . . . . . . . . . . . . . . . . 19 class 𝑓
59	51, 58	cfv 6481	. . . . . . . . . . . . . . . . . 18 class (𝑓‘𝑥)
60	32	cv 1540	. . . . . . . . . . . . . . . . . 18 class 𝑙
61	59, 53, 60	co 7346	. . . . . . . . . . . . . . . . 17 class ((𝑓‘𝑥)𝑙(𝑚‘𝑥))
62	57, 61	cfv 6481	. . . . . . . . . . . . . . . 16 class (((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))
63	29	cv 1540	. . . . . . . . . . . . . . . . . . 19 class 𝑘
64	59, 63	cfv 6481	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑓‘𝑥))
65	53, 63	cfv 6481	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑚‘𝑥))
66	64, 65	cop 4582	. . . . . . . . . . . . . . . . 17 class ⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩
67	38	cv 1540	. . . . . . . . . . . . . . . . . 18 class 𝑟
68	53, 67	cfv 6481	. . . . . . . . . . . . . . . . 17 class (𝑟‘(𝑚‘𝑥))
69		cco 17170	. . . . . . . . . . . . . . . . . 18 class comp
70	14, 69	cfv 6481	. . . . . . . . . . . . . . . . 17 class (comp‘𝑒)
71	66, 68, 70	co 7346	. . . . . . . . . . . . . . . 16 class (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))
72	55, 62, 71	co 7346	. . . . . . . . . . . . . . 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 5172	. . . . . . . . . . . . . 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 7348	. . . . . . . . . . . . 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 3850	. . . . . . . . . . . 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 3850	. . . . . . . . . . 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 3850	. . . . . . . . . 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 3850	. . . . . . . . 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 3850	. . . . . . . 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 7348	. . . . . . 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 4582	. . . . . 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 3850	. . . . 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 3850	. . . 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 3850	. . 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 7348	. 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 1541	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 referenced by: fucofvalg 49349

W3C validator