fucofvalg - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fucofvalg		Structured version Visualization version GIF version

Theorem fucofvalg 49903

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 49902	. . . 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 6878	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) ∈ V)
5		simprl 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → 𝑝 = 𝑃)
6	5	fveq2d 6867	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) = (1^st ‘𝑃))
7		fucofvalg.c	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑃) = 𝐶)
8	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑃) = 𝐶)
9	6, 8	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) → (1^st ‘𝑝) = 𝐶)
10		fvexd 6878	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑝) ∈ V)
11		simplrl 786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → 𝑝 = 𝑃)
12	11	fveq2d 6867	. . . . . 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 736	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑃) = 𝐷)
15	12, 14	eqtrd 2796	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2^nd ‘𝑝) = 𝐷)
16		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
17		simpllr 785	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑝 = 𝑃 ∧ 𝑒 = 𝐸))
18	17	simprd 499	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑒 = 𝐸)
19	16, 18	oveq12d 7410	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
20		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
21	20, 16	oveq12d 7410	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
22	19, 21	xpeq12d 5676	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
23		ovexd 7427	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝐷 Func 𝐸) ∈ V)
24		ovexd 7427	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝐶 Func 𝐷) ∈ V)
25	23, 24	xpexd 7730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ∈ V)
26	22, 25	eqeltrd 2861	. . . . . 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 740	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	22, 28	eqtr4d 2799	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) = 𝑊)
30		simpr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
31	30	reseq2d 5963	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → ( ∘_func ↾ 𝑤) = ( ∘_func ↾ 𝑊))
32		simplr 778	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑑 = 𝐷)
33	18	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑒 = 𝐸)
34	32, 33	oveq12d 7410	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (𝑑 Nat 𝑒) = (𝐷 Nat 𝐸))
35	34	oveqd 7409	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → 𝑐 = 𝐶)
37	36, 32	oveq12d 7410	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (𝑐 Nat 𝑑) = (𝐶 Nat 𝐷))
38	37	oveqd 7409	. . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (Base‘𝑐) = (Base‘𝐶))
40	33	fveq2d 6867	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) ∧ 𝑤 = 𝑊) → (comp‘𝑒) = (comp‘𝐸))
41	40	oveqd 7409	. . . . . . . . . . . . . . . 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 7409	. . . . . . . . . . . . . . 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 5186	. . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . 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 3859	. . . . . . . . . . . 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 3859	. . . . . . . . . . 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 3859	. . . . . . . . . 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 3859	. . . . . . . . 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 3859	. . . . . . . 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 7467	. . . . . . 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 4838	. . . . . 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 3889	. . . . 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 3889	. . . 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 3889	. . 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 3476	. . 3 ⊢ (𝜑 → 𝑃 ∈ V)
57		fucofvalg.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
58	57	elexd 3476	. . 3 ⊢ (𝜑 → 𝐸 ∈ V)
59		opex 5430	. . . 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 7544	. 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 2798	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 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ⦋csb 3852 ⟨cop 4587 ↦ cmpt 5180 × cxp 5643 ↾ cres 5647 ‘cfv 6517 (class class class)co 7392 ∈ cmpo 7394 1^st c1st 7964 2^nd c2nd 7965 Basecbs 17228 compcco 17281 Func cfunc 17870 ∘_func ccofu 17872 Nat cnat 17960 ∘_F cfuco 49901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-res 5657 df-iota 6473 df-fun 6519 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-fuco 49902

This theorem is referenced by: fucofval 49904 fucofvalne 49910

W3C validator