df-fuco - Mathbox for Zhi Wang

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Definition df-fuco 49819

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 49861). The object part maps two functors to their composition (fuco11 49828 and fuco11b 49839). The morphism part defines the "composition" of two natural transformations (fuco22 49841) into another natural transformation (fuco22nat 49848) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49854). Note that such "composition" is different from fucco 17927 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 49818	. 2 class ∘_F
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3433	. . 3 class V
5		vc	. . . 4 setvar 𝑐
6	2	cv 1547	. . . . 5 class 𝑝
7		c1st 7931	. . . . 5 class 1^st
8	6, 7	cfv 6488	. . . 4 class (1^st ‘𝑝)
9		vd	. . . . 5 setvar 𝑑
10		c2nd 7932	. . . . . 6 class 2^nd
11	6, 10	cfv 6488	. . . . 5 class (2^nd ‘𝑝)
12		vw	. . . . . 6 setvar 𝑤
13	9	cv 1547	. . . . . . . 8 class 𝑑
14	3	cv 1547	. . . . . . . 8 class 𝑒
15		cfunc 17816	. . . . . . . 8 class Func
16	13, 14, 15	co 7359	. . . . . . 7 class (𝑑 Func 𝑒)
17	5	cv 1547	. . . . . . . 8 class 𝑐
18	17, 13, 15	co 7359	. . . . . . 7 class (𝑐 Func 𝑑)
19	16, 18	cxp 5618	. . . . . 6 class ((𝑑 Func 𝑒) × (𝑐 Func 𝑑))
20		ccofu 17818	. . . . . . . 8 class ∘_func
21	12	cv 1547	. . . . . . . 8 class 𝑤
22	20, 21	cres 5622	. . . . . . 7 class ( ∘_func ↾ 𝑤)
23		vu	. . . . . . . 8 setvar 𝑢
24		vv	. . . . . . . 8 setvar 𝑣
25		vf	. . . . . . . . 9 setvar 𝑓
26	23	cv 1547	. . . . . . . . . . 11 class 𝑢
27	26, 10	cfv 6488	. . . . . . . . . 10 class (2^nd ‘𝑢)
28	27, 7	cfv 6488	. . . . . . . . 9 class (1^st ‘(2^nd ‘𝑢))
29		vk	. . . . . . . . . 10 setvar 𝑘
30	26, 7	cfv 6488	. . . . . . . . . . 11 class (1^st ‘𝑢)
31	30, 7	cfv 6488	. . . . . . . . . 10 class (1^st ‘(1^st ‘𝑢))
32		vl	. . . . . . . . . . 11 setvar 𝑙
33	30, 10	cfv 6488	. . . . . . . . . . 11 class (2^nd ‘(1^st ‘𝑢))
34		vm	. . . . . . . . . . . 12 setvar 𝑚
35	24	cv 1547	. . . . . . . . . . . . . 14 class 𝑣
36	35, 10	cfv 6488	. . . . . . . . . . . . 13 class (2^nd ‘𝑣)
37	36, 7	cfv 6488	. . . . . . . . . . . 12 class (1^st ‘(2^nd ‘𝑣))
38		vr	. . . . . . . . . . . . 13 setvar 𝑟
39	35, 7	cfv 6488	. . . . . . . . . . . . . 14 class (1^st ‘𝑣)
40	39, 7	cfv 6488	. . . . . . . . . . . . 13 class (1^st ‘(1^st ‘𝑣))
41		vb	. . . . . . . . . . . . . 14 setvar 𝑏
42		va	. . . . . . . . . . . . . 14 setvar 𝑎
43		cnat 17906	. . . . . . . . . . . . . . . 16 class Nat
44	13, 14, 43	co 7359	. . . . . . . . . . . . . . 15 class (𝑑 Nat 𝑒)
45	30, 39, 44	co 7359	. . . . . . . . . . . . . 14 class ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣))
46	17, 13, 43	co 7359	. . . . . . . . . . . . . . 15 class (𝑐 Nat 𝑑)
47	27, 36, 46	co 7359	. . . . . . . . . . . . . 14 class ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣))
48		vx	. . . . . . . . . . . . . . 15 setvar 𝑥
49		cbs 17174	. . . . . . . . . . . . . . . 16 class Base
50	17, 49	cfv 6488	. . . . . . . . . . . . . . 15 class (Base‘𝑐)
51	48	cv 1547	. . . . . . . . . . . . . . . . . 18 class 𝑥
52	34	cv 1547	. . . . . . . . . . . . . . . . . 18 class 𝑚
53	51, 52	cfv 6488	. . . . . . . . . . . . . . . . 17 class (𝑚‘𝑥)
54	41	cv 1547	. . . . . . . . . . . . . . . . 17 class 𝑏
55	53, 54	cfv 6488	. . . . . . . . . . . . . . . 16 class (𝑏‘(𝑚‘𝑥))
56	42	cv 1547	. . . . . . . . . . . . . . . . . 18 class 𝑎
57	51, 56	cfv 6488	. . . . . . . . . . . . . . . . 17 class (𝑎‘𝑥)
58	25	cv 1547	. . . . . . . . . . . . . . . . . . 19 class 𝑓
59	51, 58	cfv 6488	. . . . . . . . . . . . . . . . . 18 class (𝑓‘𝑥)
60	32	cv 1547	. . . . . . . . . . . . . . . . . 18 class 𝑙
61	59, 53, 60	co 7359	. . . . . . . . . . . . . . . . 17 class ((𝑓‘𝑥)𝑙(𝑚‘𝑥))
62	57, 61	cfv 6488	. . . . . . . . . . . . . . . 16 class (((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))
63	29	cv 1547	. . . . . . . . . . . . . . . . . . 19 class 𝑘
64	59, 63	cfv 6488	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑓‘𝑥))
65	53, 63	cfv 6488	. . . . . . . . . . . . . . . . . 18 class (𝑘‘(𝑚‘𝑥))
66	64, 65	cop 4563	. . . . . . . . . . . . . . . . 17 class ⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩
67	38	cv 1547	. . . . . . . . . . . . . . . . . 18 class 𝑟
68	53, 67	cfv 6488	. . . . . . . . . . . . . . . . 17 class (𝑟‘(𝑚‘𝑥))
69		cco 17227	. . . . . . . . . . . . . . . . . 18 class comp
70	14, 69	cfv 6488	. . . . . . . . . . . . . . . . 17 class (comp‘𝑒)
71	66, 68, 70	co 7359	. . . . . . . . . . . . . . . 16 class (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))
72	55, 62, 71	co 7359	. . . . . . . . . . . . . . 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 5155	. . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . 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 3832	. . . . . . . . . . . 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 3832	. . . . . . . . . . 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 3832	. . . . . . . . . 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 3832	. . . . . . . . 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 3832	. . . . . . . 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 7361	. . . . . . 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 4563	. . . . . 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 3832	. . . . 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 3832	. . . 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 3832	. . 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 7361	. 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 1548	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 49820

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