fucofvalg - Mathbox for Zhi Wang

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Theorem fucofvalg 49559

Description: Value of the function giving the functor composition bifunctor. (Contributed by Zhi Wang, 7-Oct-2025.)

**Hypotheses**
Ref	Expression
fucofvalg.p	⊢ (𝜑 → 𝑃 ∈ 𝑈)
fucofvalg.c	⊢ (𝜑 → (1^st ‘𝑃) = 𝐶)
fucofvalg.d	⊢ (𝜑 → (2^nd ‘𝑃) = 𝐷)
fucofvalg.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
fucofvalg.o	⊢ (𝜑 → (𝑃 ∘_F 𝐸) = ⚬ )
fucofvalg.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))

**Assertion**
Ref	Expression
fucofvalg	⊢ (𝜑 → ⚬ = ⟨( ∘_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 groups: 𝐶,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐷,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐸,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝑊,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝜑,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝑃,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 Allowed substitution hints: 𝑈(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙) 𝑉(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙) ⚬ (𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)

Proof of Theorem fucofvalg Dummy variables 𝑐 𝑑 𝑒 𝑝 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucofvalg.o	. 2 ⊢ (𝜑 → (𝑃 ∘_F 𝐸) = ⚬ )
2		df-fuco 49558	. . . 4 ⊢ ∘_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))))))⟩)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ∘_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))))))⟩))
4		fvexd 6849	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) ∈ V)
5		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → 𝑝 = 𝑃)
6	5	fveq2d 6838	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) = (1^st ‘𝑃))
7		fucofvalg.c	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑃) = 𝐶)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑃) = 𝐶)
9	6, 8	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) = 𝐶)
10		fvexd 6849	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑝) ∈ V)
11		simplrl 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → 𝑝 = 𝑃)
12	11	fveq2d 6838	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑝) = (2^nd ‘𝑃))
13		fucofvalg.d	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘𝑃) = 𝐷)
14	13	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑃) = 𝐷)
15	12, 14	eqtrd 2771	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑝) = 𝐷)
16		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
17		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑝 = 𝑃 ∧ 𝑒 = 𝐸))
18	17	simprd 495	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑒 = 𝐸)
19	16, 18	oveq12d 7376	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
20		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
21	20, 16	oveq12d 7376	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
22	19, 21	xpeq12d 5655	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23		ovexd 7393	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝐷 Func 𝐸) ∈ V)
24		ovexd 7393	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝐶 Func 𝐷) ∈ V)
25	23, 24	xpexd 7696	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V)
26	22, 25	eqeltrd 2836	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) ∈ V)
27		fucofvalg.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	27	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	22, 28	eqtr4d 2774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) = 𝑊)
30		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
31	30	reseq2d 5938	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ( ∘_func ↾ 𝑤) = ( ∘_func ↾ 𝑊))
32		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑑 = 𝐷)
33	18	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑒 = 𝐸)
34	32, 33	oveq12d 7376	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (𝑑 Nat 𝑒) = (𝐷 Nat 𝐸))
35	34	oveqd 7375	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)) = ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)))
36		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑐 = 𝐶)
37	36, 32	oveq12d 7376	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
38	37	oveqd 7375	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) = ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)))
39	36	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (Base‘𝑐) = (Base‘𝐶))
40	33	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (comp‘𝑒) = (comp‘𝐸))
41	40	oveqd 7375	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥))) = (⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥))))
42	41	oveqd 7375	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))) = ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))
43	39, 42	mpteq12dv 5185	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))
44	35, 38, 43	mpoeq123dv 7433	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (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))))) = (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))))))
45	44	csbeq2dv 3856	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ⦋(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))))) = ⦋(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))))))
46	45	csbeq2dv 3856	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ⦋(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))))) = ⦋(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))))))
47	46	csbeq2dv 3856	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ⦋(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))))) = ⦋(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))))))
48	47	csbeq2dv 3856	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ⦋(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))))) = ⦋(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))))))
49	48	csbeq2dv 3856	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 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))))) = ⦋(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))))))
50	30, 30, 49	mpoeq123dv 7433	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 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)))))) = (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)))))))
51	31, 50	opeq12d 4837	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 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))))))⟩ = ⟨( ∘_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))))))⟩)
52	26, 29, 51	csbied2 3886	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 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))))))⟩ = ⟨( ∘_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))))))⟩)
53	10, 15, 52	csbied2 3886	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 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))))))⟩ = ⟨( ∘_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))))))⟩)
54	4, 9, 53	csbied2 3886	. . 3 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → ⦋(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))))))⟩ = ⟨( ∘_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))))))⟩)
55		fucofvalg.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑈)
56	55	elexd 3464	. . 3 ⊢ (𝜑 → 𝑃 ∈ V)
57		fucofvalg.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
58	57	elexd 3464	. . 3 ⊢ (𝜑 → 𝐸 ∈ V)
59		opex 5412	. . . 4 ⊢ ⟨( ∘_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))))))⟩ ∈ V
60	59	a1i 11	. . 3 ⊢ (𝜑 → ⟨( ∘_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))))))⟩ ∈ V)
61	3, 54, 56, 58, 60	ovmpod 7510	. 2 ⊢ (𝜑 → (𝑃 ∘_F 𝐸) = ⟨( ∘_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))))))⟩)
62	1, 61	eqtr3d 2773	1 ⊢ (𝜑 → ⚬ = ⟨( ∘_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

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ⦋csb 3849 ⟨cop 4586 ↦ cmpt 5179 × cxp 5622 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 1^st c1st 7931 2^nd c2nd 7932 Basecbs 17136 compcco 17189 Func cfunc 17778 ∘_func ccofu 17780 Nat cnat 17868 ∘_F cfuco 49557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-res 5636 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-fuco 49558

This theorem is referenced by: fucofval 49560 fucofvalne 49566

W3C validator