df-fuco - Mathbox for Zhi Wang

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Definition df-fuco 49804

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 49846). The object part maps two functors to their composition (fuco11 49813 and fuco11b 49824). The morphism part defines the "composition" of two natural transformations (fuco22 49826) into another natural transformation (fuco22nat 49833) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49839). Note that such "composition" is different from fucco 17923 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 49803	. 2 class ∘_F
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3430	. . 3 class V
5		vc	. . . 4 setvar 𝑐
6	2	cv 1541	. . . . 5 class 𝑝
7		c1st 7933	. . . . 5 class 1^st
8	6, 7	cfv 6492	. . . 4 class (1^st ‘𝑝)
9		vd	. . . . 5 setvar 𝑑
10		c2nd 7934	. . . . . 6 class 2^nd
11	6, 10	cfv 6492	. . . . 5 class (2^nd ‘𝑝)
12		vw	. . . . . 6 setvar 𝑤
13	9	cv 1541	. . . . . . . 8 class 𝑑
14	3	cv 1541	. . . . . . . 8 class 𝑒
15		cfunc 17812	. . . . . . . 8 class Func
16	13, 14, 15	co 7360	. . . . . . 7 class (𝑑 Func 𝑒)
17	5	cv 1541	. . . . . . . 8 class 𝑐
18	17, 13, 15	co 7360	. . . . . . 7 class (𝑐 Func 𝑑)
19	16, 18	cxp 5622	. . . . . 6 class ((𝑑 Func 𝑒) × (𝑐 Func 𝑑))
20		ccofu 17814	. . . . . . . 8 class ∘_func
21	12	cv 1541	. . . . . . . 8 class 𝑤
22	20, 21	cres 5626	. . . . . . 7 class ( ∘_func ↾ 𝑤)
23		vu	. . . . . . . 8 setvar 𝑢
24		vv	. . . . . . . 8 setvar 𝑣
25		vf	. . . . . . . . 9 setvar 𝑓
26	23	cv 1541	. . . . . . . . . . 11 class 𝑢
27	26, 10	cfv 6492	. . . . . . . . . 10 class (2^nd ‘𝑢)
28	27, 7	cfv 6492	. . . . . . . . 9 class (1^st ‘(2^nd ‘𝑢))
29		vk	. . . . . . . . . 10 setvar 𝑘
30	26, 7	cfv 6492	. . . . . . . . . . 11 class (1^st ‘𝑢)
31	30, 7	cfv 6492	. . . . . . . . . 10 class (1^st ‘(1^st ‘𝑢))
32		vl	. . . . . . . . . . 11 setvar 𝑙
33	30, 10	cfv 6492	. . . . . . . . . . 11 class (2^nd ‘(1^st ‘𝑢))
34		vm	. . . . . . . . . . . 12 setvar 𝑚
35	24	cv 1541	. . . . . . . . . . . . . 14 class 𝑣
36	35, 10	cfv 6492	. . . . . . . . . . . . 13 class (2^nd ‘𝑣)
37	36, 7	cfv 6492	. . . . . . . . . . . 12 class (1^st ‘(2^nd ‘𝑣))
38		vr	. . . . . . . . . . . . 13 setvar 𝑟
39	35, 7	cfv 6492	. . . . . . . . . . . . . 14 class (1^st ‘𝑣)
40	39, 7	cfv 6492	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑣))
41		vb	. . . . . . . . . . . . . 14 setvar 𝑏
42		va	. . . . . . . . . . . . . 14 setvar 𝑎
43		cnat 17902	. . . . . . . . . . . . . . . 16 class Nat
44	13, 14, 43	co 7360	. . . . . . . . . . . . . . 15 class (𝑑 Nat 𝑒)
45	30, 39, 44	co 7360	. . . . . . . . . . . . . 14 class ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣))
46	17, 13, 43	co 7360	. . . . . . . . . . . . . . 15 class (𝑐 Nat 𝑑)
47	27, 36, 46	co 7360	. . . . . . . . . . . . . 14 class ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣))
48		vx	. . . . . . . . . . . . . . 15 setvar 𝑥
49		cbs 17170	. . . . . . . . . . . . . . . 16 class Base
50	17, 49	cfv 6492	. . . . . . . . . . . . . . 15 class (Base‘𝑐)
51	48	cv 1541	. . . . . . . . . . . . . . . . . 18 class 𝑥
52	34	cv 1541	. . . . . . . . . . . . . . . . . 18 class 𝑚
53	51, 52	cfv 6492	. . . . . . . . . . . . . . . . 17 class (𝑚‘𝑥)
54	41	cv 1541	. . . . . . . . . . . . . . . . 17 class 𝑏
55	53, 54	cfv 6492	. . . . . . . . . . . . . . . 16 class (𝑏‘(𝑚‘𝑥))
56	42	cv 1541	. . . . . . . . . . . . . . . . . 18 class 𝑎
57	51, 56	cfv 6492	. . . . . . . . . . . . . . . . 17 class (𝑎‘𝑥)
58	25	cv 1541	. . . . . . . . . . . . . . . . . . 19 class 𝑓
59	51, 58	cfv 6492	. . . . . . . . . . . . . . . . . 18 class (𝑓‘𝑥)
60	32	cv 1541	. . . . . . . . . . . . . . . . . 18 class 𝑙
61	59, 53, 60	co 7360	. . . . . . . . . . . . . . . . 17 class ((𝑓‘𝑥)𝑙(𝑚‘𝑥))
62	57, 61	cfv 6492	. . . . . . . . . . . . . . . 16 class (((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))
63	29	cv 1541	. . . . . . . . . . . . . . . . . . 19 class 𝑘
64	59, 63	cfv 6492	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑓‘𝑥))
65	53, 63	cfv 6492	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑚‘𝑥))
66	64, 65	cop 4574	. . . . . . . . . . . . . . . . 17 class ⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩
67	38	cv 1541	. . . . . . . . . . . . . . . . . 18 class 𝑟
68	53, 67	cfv 6492	. . . . . . . . . . . . . . . . 17 class (𝑟‘(𝑚‘𝑥))
69		cco 17223	. . . . . . . . . . . . . . . . . 18 class comp
70	14, 69	cfv 6492	. . . . . . . . . . . . . . . . 17 class (comp‘𝑒)
71	66, 68, 70	co 7360	. . . . . . . . . . . . . . . 16 class (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))
72	55, 62, 71	co 7360	. . . . . . . . . . . . . . 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 5167	. . . . . . . . . . . . . 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 7362	. . . . . . . . . . . . 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 3838	. . . . . . . . . . . 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 3838	. . . . . . . . . . 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 3838	. . . . . . . . . 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 3838	. . . . . . . . 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 3838	. . . . . . . 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 7362	. . . . . . 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 4574	. . . . . 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 3838	. . . . 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 3838	. . . 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 3838	. . 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 7362	. 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 1542	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 49805

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