Theorem fucofvalne 50540

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 5260	. . . 4 ⊢ ∅ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
3		1st0 8007	. . . 4 ⊢ (1st ‘∅) = ∅
4	3	a1i 11	. . 3 ⊢ (𝜑 → (1st ‘∅) = ∅)
5		2nd0 8008	. . . 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 4853	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = ∅)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ = ∅)
11	10	oveq1d 7435	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (∅ ∘F 𝐸))
12		fucofvalne.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )
13	11, 12	eqtr3d 2797	. . 3 ⊢ (𝜑 → (∅ ∘F 𝐸) = ⚬ )
14		eqidd 2761	. . 3 ⊢ (𝜑 → ((∅ Func 𝐸) × (∅ Func ∅)) = ((∅ Func 𝐸) × (∅ Func ∅)))
15	2, 4, 6, 7, 13, 14	fucofvalg 50533	. 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 5431	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ∈ V
17	16	snnz 4733	. . . . . . . . 9 ⊢ {⟨∅, ∅⟩} ≠ ∅
18	17	neii 2957	. . . . . . . 8 ⊢ ¬ {⟨∅, ∅⟩} = ∅
19		ioran 999	. . . . . . . . 9 ⊢ (¬ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅) ↔ (¬ {⟨∅, ∅⟩} = ∅ ∧ ¬ {⟨∅, ∅⟩} = ∅))
20		xpeq0 6150	. . . . . . . . . . 11 ⊢ (({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅ ↔ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅))
21	20	biimpi 219	. . . . . . . . . 10 ⊢ (({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅ → ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅))
22	21	con3i 155	. . . . . . . . 9 ⊢ (¬ ({⟨∅, ∅⟩} = ∅ ∨ {⟨∅, ∅⟩} = ∅) → ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅)
23	19, 22	sylbir 238	. . . . . . . 8 ⊢ ((¬ {⟨∅, ∅⟩} = ∅ ∧ ¬ {⟨∅, ∅⟩} = ∅) → ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅)
24	18, 18, 23	mp2an 705	. . . . . . 7 ⊢ ¬ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅
25	7	0func 50295	. . . . . . . . 9 ⊢ (𝜑 → (∅ Func 𝐸) = {⟨∅, ∅⟩})
26		0cat 17868	. . . . . . . . . . 11 ⊢ ∅ ∈ Cat
27	26	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ Cat)
28	27	0func 50295	. . . . . . . . 9 ⊢ (𝜑 → (∅ Func ∅) = {⟨∅, ∅⟩})
29	25, 28	xpeq12d 5681	. . . . . . . 8 ⊢ (𝜑 → ((∅ Func 𝐸) × (∅ Func ∅)) = ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}))
30		df-func 18038	. . . . . . . . . . . 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 7554	. . . . . . . . . . 11 ⊢ Rel dom Func
32		0nelrel0 5710	. . . . . . . . . . 11 ⊢ (Rel dom Func → ¬ ∅ ∈ dom Func )
33	31, 32	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ dom Func
34	10	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ dom Func ↔ ∅ ∈ dom Func ))
35	33, 34	mtbiri 330	. . . . . . . . 9 ⊢ (𝜑 → ¬ ⟨𝐶, 𝐷⟩ ∈ dom Func )
36		df-ov 7423	. . . . . . . . . . . 12 ⊢ (𝐶 Func 𝐷) = ( Func ‘⟨𝐶, 𝐷⟩)
37		ndmfv 6917	. . . . . . . . . . . 12 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ( Func ‘⟨𝐶, 𝐷⟩) = ∅)
38	36, 37	eqtrid 2807	. . . . . . . . . . 11 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → (𝐶 Func 𝐷) = ∅)
39	38	xpeq2d 5680	. . . . . . . . . 10 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × ∅))
40		xp0 5750	. . . . . . . . . 10 ⊢ ((𝐷 Func 𝐸) × ∅) = ∅
41	39, 40	eqtrdi 2811	. . . . . . . . 9 ⊢ (¬ ⟨𝐶, 𝐷⟩ ∈ dom Func → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ∅)
42	35, 41	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ∅)
43	29, 42	eqeq12d 2776	. . . . . . 7 ⊢ (𝜑 → (((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) ↔ ({⟨∅, ∅⟩} × {⟨∅, ∅⟩}) = ∅))
44	24, 43	mtbiri 330	. . . . . 6 ⊢ (𝜑 → ¬ ((∅ Func 𝐸) × (∅ Func ∅)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
45		rescofuf 50301	. . . . . . . . . 10 ⊢ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))):((∅ Func 𝐸) × (∅ Func ∅))⟶(∅ Func 𝐸)
46	45	fdmi 6721	. . . . . . . . 9 ⊢ dom ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ((∅ Func 𝐸) × (∅ Func ∅))
47		rescofuf 50301	. . . . . . . . . 10 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)
48	47	fdmi 6721	. . . . . . . . 9 ⊢ dom ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
49	46, 48	eqeq12i 2778	. . . . . . . 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 219	. . . . . . 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 155	. . . . . 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 5884	. . . . . . 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 155	. . . . . 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 19	. . . . 5 ⊢ (𝜑 → ¬ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
55	54	neqned 2962	. . . 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 5969	. . . 4 ⊢ (𝜑 → ( ∘func ↾ 𝑊) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
58	55, 57	neeqtrrd 3029	. . 3 ⊢ (𝜑 → ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ≠ ( ∘func ↾ 𝑊))
59		ovex 7453	. . . . . 6 ⊢ (∅ Func 𝐸) ∈ V
60		ovex 7453	. . . . . 6 ⊢ (∅ Func ∅) ∈ V
61	59, 60	xpex 7767	. . . . 5 ⊢ ((∅ Func 𝐸) × (∅ Func ∅)) ∈ V
62		fex 7232	. . . . 5 ⊢ ((( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))):((∅ Func 𝐸) × (∅ Func ∅))⟶(∅ Func 𝐸) ∧ ((∅ Func 𝐸) × (∅ Func ∅)) ∈ V) → ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ∈ V)
63	45, 61, 62	mp2an 705	. . . 4 ⊢ ( ∘func ↾ ((∅ Func 𝐸) × (∅ Func ∅))) ∈ V
64	61, 61	mpoex 8092	. . . 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 50030	. . . 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 705	. . 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 18	. 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 3022	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))))))⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  [wsbc 3738  ⦋csb 3846  ∅c0 4278  {csn 4584  ⟨cop 4834  {copab 5166   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   ↾ cres 5652  Rel wrel 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8850  Xcixp 8928  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Idccid 17844   Func cfunc 18034   ∘_func ccofu 18036   Nat cnat 18124   ∘_F cfuco 50531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-1cn 11261  ax-addcl 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-map 8852  df-ixp 8929  df-nn 12339  df-slot 17365  df-ndx 17377  df-base 17393  df-cat 17847  df-cid 17848  df-func 18038  df-cofu 18040  df-fuco 50532

This theorem is used by: (None)