Theorem fuconuniqlem 50872

Description: Lemma for fuconuniq 50873. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fuconuniq2.c	⊢ 𝐶 = (SetCat‘1o)
fuconuniq2.e	⊢ 𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}
fuconuniq2.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fuconuniq2.l	⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
fuconuniq.a	⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
fuconuniq.m	⊢ 𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
fuconuniq.s	⊢ 𝑆 = ⟨( I ↾ 2o), 𝑀⟩
fuconuniq.r	⊢ 𝑅 = (𝐸 FuncCat 𝐸)
fuconuniq.o	⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
fuconuniq.k	⊢ 𝐾 = ((1st ‘ ⚬ )‘𝑆)
fuconuniqlem.t	⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))
fuconuniqlem.d	⊢ 𝐷 = (𝐸Δfunc𝐶)
fuconuniqlem.f	⊢ 𝐹 = ((1st ‘𝐷)‘∅)
fuconuniqlem.g	⊢ 𝐺 = ((1st ‘𝐷)‘1o)
fuconuniqlem.n	⊢ 𝑁 = (1o × {∅})

Assertion

Ref	Expression
fuconuniqlem	⊢ ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)) ∧ (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))

Distinct variable groups:   𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑢,𝑣,𝑎,𝑏   𝐴,𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑢,𝑣,𝑎

Allowed substitution hints:   𝐴(𝑏)   𝐶(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐷(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑄(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑅(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑆(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑇(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐸(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐹(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐺(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐾(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝐿(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑀(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   𝑁(𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)   ⚬ (𝑣, 𝑢, 𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏)

Proof of Theorem fuconuniqlem

Dummy variables 𝑥 𝑦 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuconuniq2.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
2		fuconuniqlem.t	. . . 4 ⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))
3		fuconuniq2.l	. . . 4 ⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
4		fuconuniq2.c	. . . . . . 7 ⊢ 𝐶 = (SetCat‘1o)
5		setc1oterm 50717	. . . . . . 7 ⊢ (SetCat‘1o) ∈ TermCat
6	4, 5	eqeltri 2856	. . . . . 6 ⊢ 𝐶 ∈ TermCat
7	6	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ TermCat)
8	7	termccatd 50705	. . . 4 ⊢ (⊤ → 𝐶 ∈ Cat)
9		fuconuniq2.e	. . . . . . 7 ⊢ 𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}
10	9	par2catid 50858	. . . . . 6 ⊢ (𝐸 ∈ Cat ∧ (Id‘𝐸) = (𝑦 ∈ 2o ↦ 2o))
11	10	simpli 489	. . . . 5 ⊢ 𝐸 ∈ Cat
12	11	a1i 11	. . . 4 ⊢ (⊤ → 𝐸 ∈ Cat)
13		fuconuniq.r	. . . . . 6 ⊢ 𝑅 = (𝐸 FuncCat 𝐸)
14		fuconuniq.o	. . . . . 6 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
15		fuconuniq.k	. . . . . 6 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝑆)
16		fuconuniq.s	. . . . . . . 8 ⊢ 𝑆 = ⟨( I ↾ 2o), 𝑀⟩
17		fuconuniq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
18		fuconuniq.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
19	9, 17, 18	parswapfunc 50869	. . . . . . . 8 ⊢ ⟨( I ↾ 2o), 𝑀⟩ ∈ (𝐸 Func 𝐸)
20	16, 19	eqeltri 2856	. . . . . . 7 ⊢ 𝑆 ∈ (𝐸 Func 𝐸)
21	20	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑆 ∈ (𝐸 Func 𝐸))
22		mptresid 6042	. . . . . . . . 9 ⊢ ( I ↾ 2o) = (𝑥 ∈ 2o ↦ 𝑥)
23		2oex 8490	. . . . . . . . . 10 ⊢ 2o ∈ V
24	23	mptex 7229	. . . . . . . . 9 ⊢ (𝑥 ∈ 2o ↦ 𝑥) ∈ V
25	22, 24	eqeltri 2856	. . . . . . . 8 ⊢ ( I ↾ 2o) ∈ V
26	23, 23	mpoex 8092	. . . . . . . . 9 ⊢ (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎))) ∈ V
27	18, 26	eqeltri 2856	. . . . . . . 8 ⊢ 𝑀 ∈ V
28	25, 27	op1std 8011	. . . . . . 7 ⊢ (𝑆 = ⟨( I ↾ 2o), 𝑀⟩ → (1st ‘𝑆) = ( I ↾ 2o))
29	16, 28	mp1i 14	. . . . . 6 ⊢ (⊤ → (1st ‘𝑆) = ( I ↾ 2o))
30	7, 1, 13, 14, 15, 21, 29	termcpostfix 50763	. . . . 5 ⊢ (⊤ → (𝐾 ∈ (𝑄 Func 𝑄) ∧ (1st ‘𝐾) = ( I ↾ (𝐶 Func 𝐸))))
31	30	simpld 500	. . . 4 ⊢ (⊤ → 𝐾 ∈ (𝑄 Func 𝑄))
32	30	simprd 501	. . . 4 ⊢ (⊤ → (1st ‘𝐾) = ( I ↾ (𝐶 Func 𝐸)))
33	1, 2, 3, 8, 8, 12, 31, 32	fucoaltfunc 50582	. . 3 ⊢ (⊤ → (1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)))
34		fuconuniqlem.d	. . . . . . . 8 ⊢ 𝐷 = (𝐸Δfunc𝐶)
35		fuconuniqlem.f	. . . . . . . 8 ⊢ 𝐹 = ((1st ‘𝐷)‘∅)
36		fuconuniqlem.g	. . . . . . . 8 ⊢ 𝐺 = ((1st ‘𝐷)‘1o)
37		fuconuniqlem.n	. . . . . . . 8 ⊢ 𝑁 = (1o × {∅})
38	4, 9, 34, 35, 36, 37	par2diag 50871	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝑁‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))
39	38	simp1i 1157	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸))
40	39	simpli 489	. . . . 5 ⊢ 𝐹 ∈ (𝐶 Func 𝐸)
41	40	a1i 11	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐶 Func 𝐸))
42	39	simpri 491	. . . . 5 ⊢ 𝐺 ∈ (𝐶 Func 𝐸)
43	42	a1i 11	. . . 4 ⊢ (⊤ → 𝐺 ∈ (𝐶 Func 𝐸))
44	38	simp2i 1158	. . . . . 6 ⊢ (𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝑁‘∅) = ∅)
45	44	simpli 489	. . . . 5 ⊢ 𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺)
46	45	a1i 11	. . . 4 ⊢ (⊤ → 𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
47		0lt1o 8514	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 1o
48	4	setc1obas 50718	. . . . . . . . . . . . 13 ⊢ 1o = (Base‘𝐶)
49	47, 48	eleqtri 2858	. . . . . . . . . . . 12 ⊢ ∅ ∈ (Base‘𝐶)
50	49	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ∅ ∈ (Base‘𝐶))
51	1, 13, 14, 21, 8, 15, 41, 43, 46, 50	postcof2 50591	. . . . . . . . . 10 ⊢ (⊤ → (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)))
52	51	mptru 1576	. . . . . . . . 9 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅))
53	38	simp3i 1159	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o)
54	53	simpli 489	. . . . . . . . . . 11 ⊢ ((1st ‘𝐹)‘∅) = ∅
55	53	simpri 491	. . . . . . . . . . 11 ⊢ ((1st ‘𝐺)‘∅) = 1o
56	25, 27	op2ndd 8012	. . . . . . . . . . . 12 ⊢ (𝑆 = ⟨( I ↾ 2o), 𝑀⟩ → (2nd ‘𝑆) = 𝑀)
57	16, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ (2nd ‘𝑆) = 𝑀
58	54, 55, 57	oveq123i 7434	. . . . . . . . . 10 ⊢ (((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅)) = (∅𝑀1o)
59	44	simpri 491	. . . . . . . . . 10 ⊢ (𝑁‘∅) = ∅
60	58, 59	fveq12i 6891	. . . . . . . . 9 ⊢ ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)) = ((∅𝑀1o)‘∅)
61	52, 60	eqtri 2783	. . . . . . . 8 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((∅𝑀1o)‘∅)
62		0ex 5260	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
63	62	prid1 4722	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
64		df2o3 8486	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
65	63, 64	eleqtrri 2859	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
66	65	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ 2o)
67		1oelpr 8489	. . . . . . . . . . . 12 ⊢ 1o ∈ {∅, 1o}
68	67, 64	eleqtrri 2859	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
69	68	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 1o ∈ 2o)
70		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅))) = (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))
71	70	par2hom 50870	. . . . . . . . . . . . . 14 ⊢ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = {2o} ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = {2o} ∧ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = ∅))
72	71	simp3i 1159	. . . . . . . . . . . . 13 ⊢ ((∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o ∧ (1o(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))∅) = ∅)
73	72	simpli 489	. . . . . . . . . . . 12 ⊢ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o
74	65, 73	eleqtrri 2859	. . . . . . . . . . 11 ⊢ ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o)
75	74	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o))
76	18, 66, 69, 75	fvmpomptd 50074	. . . . . . . . 9 ⊢ (⊤ → ((∅𝑀1o)‘∅) = (𝐴‘∅))
77	76	mptru 1576	. . . . . . . 8 ⊢ ((∅𝑀1o)‘∅) = (𝐴‘∅)
78	17	parswapprop 50868	. . . . . . . . 9 ⊢ ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)
79	78	simp3i 1159	. . . . . . . 8 ⊢ (𝐴‘∅) = 1o
80	61, 77, 79	3eqtri 2787	. . . . . . 7 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = 1o
81		1n0 8497	. . . . . . 7 ⊢ 1o ≠ ∅
82	80, 81	eqnetri 3025	. . . . . 6 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) ≠ ∅
83	82, 59	neeqtrri 3028	. . . . 5 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) ≠ (𝑁‘∅)
84		fveq1 6884	. . . . . 6 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁) = 𝑁 → (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = (𝑁‘∅))
85	84	necon3i 2987	. . . . 5 ⊢ ((((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) ≠ (𝑁‘∅) → ((𝐹(2nd ‘𝐾)𝐺)‘𝑁) ≠ 𝑁)
86	83, 85	mp1i 14	. . . 4 ⊢ (⊤ → ((𝐹(2nd ‘𝐾)𝐺)‘𝑁) ≠ 𝑁)
87	1, 3, 8, 12, 31, 41, 43, 46, 86	fucoaltne 50586	. . 3 ⊢ (⊤ → (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))
88	33, 87	jca 521	. 2 ⊢ (⊤ → ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)) ∧ (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿)))
89	88	mptru 1576	1 ⊢ ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)) ∧ (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ∩ cin 4089  ∅c0 4278  ifcif 4481  {csn 4583  {cpr 4585  {ctp 4587  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   × cxp 5648   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  1_oc1o 8471  2_oc2o 8472  3_oc3o 8473  ndxcnx 17373  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841   Func cfunc 18031   ∘_func ccofu 18033   Nat cnat 18121   FuncCat cfuc 18122  SetCatcsetc 18252   ×_c cxpc 18344   curry_F ccurf 18386  Δ_funccdiag 18388   ∘_F cfuco 50529  TermCatctermc 50698

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 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

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-nel 3062  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 4584  df-pr 4586  df-tp 4588  df-op 4590  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 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-2o 8479  df-3o 8480  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-idfu 18036  df-cofu 18037  df-nat 18123  df-fuc 18124  df-setc 18253  df-xpc 18348  df-1stf 18349  df-curf 18390  df-diag 18392  df-fuco 50530  df-thinc 50644  df-termc 50699

This theorem is used by:  fuconuniq  50873