Theorem fucofvalne 48980

Description: Value of the function giving the functor composition bifunctor, if 𝐶 or 𝐷 are not sets. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
fucofvalne.c	⊢ (𝜑 → ¬ (𝐶 ∈ V ∧ 𝐷 ∈ V))
fucofvalne.e	⊢ (𝜑 → 𝐸 ∈ Cat)
fucofvalne.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )
fucofvalne.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))

Assertion

Ref	Expression
fucofvalne	⊢ (𝜑 → ⚬ ≠ ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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:   𝐸,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥   𝜑,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥

Allowed substitution hints:   𝐶(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)   𝐷(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)   𝑊(𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)   ⚬ (𝑥,𝑣,𝑢,𝑓,𝑘,𝑚,𝑟,𝑎,𝑏,𝑙)

Proof of Theorem fucofvalne

Dummy variables 𝑔 𝑛 𝑡 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5289	. . . 4 ⊢ ∅ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
3		1st0 8003	. . . 4 ⊢ (1st ‘∅) = ∅
4	3	a1i 11	. . 3 ⊢ (𝜑 → (1st ‘∅) = ∅)
5		2nd0 8004	. . . 4 ⊢ (2nd ‘∅) = ∅
6	5	a1i 11	. . 3 ⊢ (𝜑 → (2nd ‘∅) = ∅)
7		fucofvalne.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		fucofvalne.c	. . . . . 6 ⊢ (𝜑 → ¬ (𝐶 ∈ V ∧ 𝐷 ∈ V))
9		opprc 4878	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = ∅)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ = ∅)
11	10	oveq1d 7429	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (∅ ∘F 𝐸))
12		fucofvalne.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )
13	11, 12	eqtr3d 2771	. . 3 ⊢ (𝜑 → (∅ ∘F 𝐸) = ⚬ )
14		eqidd 2735	. . 3 ⊢ (𝜑 → ((∅ Func 𝐸) × (∅ Func ∅)) = ((∅ Func 𝐸) × (∅ Func ∅)))
15	2, 4, 6, 7, 13, 14	fucofvalg 48973	. 2 ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))), (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
16		opex 5451	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ∈ V
17	16	snnz 4758	. . . . . . . . 9 ⊢ {⟨∅, ∅⟩} ≠ ∅
18	17	neii 2933	. . . . . . . 8 ⊢ ¬ {⟨∅, ∅⟩} = ∅
19		ioran 985	. . . . . . . . 9 ⊢ (¬ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅) ↔ (¬ {⟨∅, ∅⟩} = ∅ ∧ ¬ {⟨∅, ∅⟩} = ∅))
20		xpeq0 6162	. . . . . . . . . . 11 ⊢ (({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅ ↔ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅))
21	20	biimpi 216	. . . . . . . . . 10 ⊢ (({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅ → ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅))
22	21	con3i 154	. . . . . . . . 9 ⊢ (¬ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅) → ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅)
23	19, 22	sylbir 235	. . . . . . . 8 ⊢ ((¬ {⟨∅, ∅⟩} = ∅ ∧ ¬ {⟨∅, ∅⟩} = ∅) → ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅)
24	18, 18, 23	mp2an 692	. . . . . . 7 ⊢ ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅
25	7	0func 48873	. . . . . . . . 9 ⊢ (𝜑 → (∅ Func 𝐸) = {⟨∅, ∅⟩})
26		0cat 17704	. . . . . . . . . . 11 ⊢ ∅ ∈ Cat
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ Cat)
28	27	0func 48873	. . . . . . . . 9 ⊢ (𝜑 → (∅ Func ∅) = {⟨∅, ∅⟩})
29	25, 28	xpeq12d 5698	. . . . . . . 8 ⊢ (𝜑 → ((∅ Func 𝐸) × (∅ Func ∅)) = ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}))
30		df-func 17875	. . . . . . . . . . . 12 ⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
31	30	reldmmpo 7550	. . . . . . . . . . 11 ⊢ Rel dom Func
32		0nelrel0 5727	. . . . . . . . . . 11 ⊢ (Rel dom Func → ¬ ∅ ∈ dom Func )
33	31, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ dom Func
34	10	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ dom Func ↔ ∅ ∈ dom Func ))
35	33, 34	mtbiri 327	. . . . . . . . 9 ⊢ (𝜑 → ¬ ⟨𝐶, 𝐷⟩ ∈ dom Func )
36		df-ov 7417	. . . . . . . . . . . 12 ⊢ (𝐶 Func 𝐷) = ( Func ‘⟨𝐶, 𝐷⟩)
37		ndmfv 6922	. . . . . . . . . . . 12 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ( Func ‘⟨𝐶, 𝐷⟩) = ∅)
38	36, 37	eqtrid 2781	. . . . . . . . . . 11 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → (𝐶 Func 𝐷) = ∅)
39	38	xpeq2d 5697	. . . . . . . . . 10 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × ∅))
40		xp0 6160	. . . . . . . . . 10 ⊢ ((𝐷 Func 𝐸) × ∅) = ∅
41	39, 40	eqtrdi 2785	. . . . . . . . 9 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ∅)
42	35, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ∅)
43	29, 42	eqeq12d 2750	. . . . . . 7 ⊢ (𝜑 → (((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ↔ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅))
44	24, 43	mtbiri 327	. . . . . 6 ⊢ (𝜑 → ¬ ((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
45		rescofuf 48877	. . . . . . . . . 10 ⊢ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))):((∅ Func 𝐸) × (∅ Func ∅))⟶(∅ Func 𝐸)
46	45	fdmi 6728	. . . . . . . . 9 ⊢ dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ((∅ Func 𝐸) × (∅ Func ∅))
47		rescofuf 48877	. . . . . . . . . 10 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)
48	47	fdmi 6728	. . . . . . . . 9 ⊢ dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
49	46, 48	eqeq12i 2752	. . . . . . . 8 ⊢ (dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ↔ ((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
50	49	biimpi 216	. . . . . . 7 ⊢ (dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
51	50	con3i 154	. . . . . 6 ⊢ (¬ ((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) → ¬ dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
52		dmeq 5896	. . . . . . 7 ⊢ (( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
53	52	con3i 154	. . . . . 6 ⊢ (¬ dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) → ¬ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
54	44, 51, 53	3syl 18	. . . . 5 ⊢ (𝜑 → ¬ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
55	54	neqned 2938	. . . 4 ⊢ (𝜑 → ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ≠ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
56		fucofvalne.w	. . . . 5 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
57	56	reseq2d 5979	. . . 4 ⊢ (𝜑 → ( ∘func ↾ 𝑊) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
58	55, 57	neeqtrrd 3005	. . 3 ⊢ (𝜑 → ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ≠ ( ∘func ↾ 𝑊))
59		ovex 7447	. . . . . 6 ⊢ (∅ Func 𝐸) ∈ V
60		ovex 7447	. . . . . 6 ⊢ (∅ Func ∅) ∈ V
61	59, 60	xpex 7756	. . . . 5 ⊢ ((∅ Func 𝐸) × (∅ Func ∅)) ∈ V
62		fex 7229	. . . . 5 ⊢ ((( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))):((∅ Func 𝐸) × (∅ Func ∅))⟶(∅ Func 𝐸) ∧ ((∅ Func 𝐸) × (∅ Func ∅)) ∈ V) → ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ∈ V)
63	45, 61, 62	mp2an 692	. . . 4 ⊢ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ∈ V
64	61, 61	mpoex 8087	. . . 4 ⊢ (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))) ∈ V
65		opth1neg 48681	. . . 4 ⊢ ((( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ∈ V ∧ (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))) ∈ V) → (( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ≠ ( ∘func ↾ 𝑊) → ⟨( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))), (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((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 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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))))))⟩))
66	63, 64, 65	mp2an 692	. . 3 ⊢ (( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ≠ ( ∘func ↾ 𝑊) → ⟨( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))), (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((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 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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))))))⟩)
67	58, 66	syl 17	. 2 ⊢ (𝜑 → ⟨( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))), (𝑢 ∈ ((∅ Func 𝐸) × (∅ Func ∅)), 𝑣 ∈ ((∅ Func 𝐸) × (∅ Func ∅)) ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(∅ Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(∅ Nat ∅)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘∅) ↦ ((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 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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))))))⟩)
68	15, 67	eqnetrd 2998	1 ⊢ (𝜑 → ⚬ ≠ ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘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))))))⟩)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  Vcvv 3464  [wsbc 3772  ⦋csb 3881  ∅c0 4315  {csn 4608  ⟨cop 4614  {copab 5187   ↦ cmpt 5207   × cxp 5665  dom cdm 5667   ↾ cres 5669  Rel wrel 5672  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7995  2^nd c2nd 7996   ↑_m cmap 8849  Xcixp 8920  Basecbs 17230  Hom chom 17285  compcco 17286  Catccat 17679  Idccid 17680   Func cfunc 17871   ∘_func ccofu 17873   Nat cnat 17961   ∘_F cfuco 48971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-map 8851  df-ixp 8921  df-nn 12250  df-slot 17202  df-ndx 17214  df-base 17231  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-fuco 48972

This theorem is referenced by: (None)