Theorem fuconuniqvalslem 50988

Description: Lemma for fuconuniqvals 50989. (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)
fuconuniqvals.d	⊢ 𝐷 = (𝐸Δfunc𝐶)
fuconuniqvals.f	⊢ 𝐹 = ((1st ‘𝐷)‘∅)
fuconuniqvals.g	⊢ 𝐺 = ((1st ‘𝐷)‘1o)
fuconuniqvals.n	⊢ 𝑁 = (1o × {∅})
fuconuniqvalslem.t	⊢ 𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))

Assertion

Ref	Expression
fuconuniqvalslem	⊢ (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)

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 fuconuniqvalslem

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

Step	Hyp	Ref	Expression
1		fuconuniq2.l	. . . . . . . . 9 ⊢ 𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)
2	1	eqcomi 2772	. . . . . . . 8 ⊢ (⟨𝐶, 𝐶⟩ ∘F 𝐸) = 𝐿
3	2	fveq2i 6911	. . . . . . 7 ⊢ (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿)
4	3	a1i 11	. . . . . 6 ⊢ (⊤ → (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿))
5		eqid 2763	. . . . . 6 ⊢ (𝐶 FuncCat 𝐶) = (𝐶 FuncCat 𝐶)
6		eqid 2763	. . . . . 6 ⊢ (Id‘(𝐶 FuncCat 𝐶)) = (Id‘(𝐶 FuncCat 𝐶))
7		eqid 2763	. . . . . 6 ⊢ (idfunc‘𝐶) = (idfunc‘𝐶)
8		fuconuniq2.c	. . . . . . . . 9 ⊢ 𝐶 = (SetCat‘1o)
9		setc1oterm 50831	. . . . . . . . 9 ⊢ (SetCat‘1o) ∈ TermCat
10	8, 9	eqeltri 2860	. . . . . . . 8 ⊢ 𝐶 ∈ TermCat
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐶 ∈ TermCat)
12	11	termccatd 50819	. . . . . 6 ⊢ (⊤ → 𝐶 ∈ Cat)
13		fuconuniq2.e	. . . . . . . . . 10 ⊢ 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)))⟩}
14		fuconuniqvals.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐸Δfunc𝐶)
15		fuconuniqvals.f	. . . . . . . . . 10 ⊢ 𝐹 = ((1st ‘𝐷)‘∅)
16		fuconuniqvals.g	. . . . . . . . . 10 ⊢ 𝐺 = ((1st ‘𝐷)‘1o)
17		fuconuniqvals.n	. . . . . . . . . 10 ⊢ 𝑁 = (1o × {∅})
18	8, 13, 14, 15, 16, 17	par2diag 50985	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝑁‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))
19	18	simp2i 1158	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝑁‘∅) = ∅)
20	19	simpli 489	. . . . . . 7 ⊢ 𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺)
21	20	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑁 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))
22	4, 5, 6, 7, 12, 21	fucoridid 50691	. . . . 5 ⊢ (⊤ → (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = 𝑁)
23	22	mptru 1577	. . . 4 ⊢ (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = 𝑁
24	23	fveq1i 6909	. . 3 ⊢ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = (𝑁‘∅)
25	19	simpri 491	. . 3 ⊢ (𝑁‘∅) = ∅
26	24, 25	eqtri 2786	. 2 ⊢ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅
27		fuconuniq2.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
28	13	par2catid 50972	. . . . . . . 8 ⊢ (𝐸 ∈ Cat ∧ (Id‘𝐸) = (𝑦 ∈ 2o ↦ 2o))
29	28	simpli 489	. . . . . . 7 ⊢ 𝐸 ∈ Cat
30	29	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐸 ∈ Cat)
31		fuconuniq.r	. . . . . . . 8 ⊢ 𝑅 = (𝐸 FuncCat 𝐸)
32		fuconuniq.o	. . . . . . . 8 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))
33		fuconuniq.k	. . . . . . . 8 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝑆)
34		fuconuniq.s	. . . . . . . . . 10 ⊢ 𝑆 = ⟨( I ↾ 2o), 𝑀⟩
35		fuconuniq.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
36		fuconuniq.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))
37	13, 35, 36	parswapfunc 50983	. . . . . . . . . 10 ⊢ ⟨( I ↾ 2o), 𝑀⟩ ∈ (𝐸 Func 𝐸)
38	34, 37	eqeltri 2860	. . . . . . . . 9 ⊢ 𝑆 ∈ (𝐸 Func 𝐸)
39	38	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝑆 ∈ (𝐸 Func 𝐸))
40		mptresid 6055	. . . . . . . . . . 11 ⊢ ( I ↾ 2o) = (𝑥 ∈ 2o ↦ 𝑥)
41		2oex 8516	. . . . . . . . . . . 12 ⊢ 2o ∈ V
42	41	mptex 7252	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 2o ↦ 𝑥) ∈ V
43	40, 42	eqeltri 2860	. . . . . . . . . 10 ⊢ ( I ↾ 2o) ∈ V
44	41, 41	mpoex 8117	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎))) ∈ V
45	36, 44	eqeltri 2860	. . . . . . . . . 10 ⊢ 𝑀 ∈ V
46	43, 45	op1std 8035	. . . . . . . . 9 ⊢ (𝑆 = ⟨( I ↾ 2o), 𝑀⟩ → (1st ‘𝑆) = ( I ↾ 2o))
47	34, 46	mp1i 14	. . . . . . . 8 ⊢ (⊤ → (1st ‘𝑆) = ( I ↾ 2o))
48	11, 27, 31, 32, 33, 39, 47	termcpostfix 50877	. . . . . . 7 ⊢ (⊤ → (𝐾 ∈ (𝑄 Func 𝑄) ∧ (1st ‘𝐾) = ( I ↾ (𝐶 Func 𝐸))))
49	48	simpld 500	. . . . . 6 ⊢ (⊤ → 𝐾 ∈ (𝑄 Func 𝑄))
50	18	simp1i 1157	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸))
51	50	simpli 489	. . . . . . 7 ⊢ 𝐹 ∈ (𝐶 Func 𝐸)
52	51	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐹 ∈ (𝐶 Func 𝐸))
53	50	simpri 491	. . . . . . 7 ⊢ 𝐺 ∈ (𝐶 Func 𝐸)
54	53	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐺 ∈ (𝐶 Func 𝐸))
55	27, 1, 12, 30, 49, 52, 54, 21, 7, 6	fucoaltval 50698	. . . . 5 ⊢ (⊤ → (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = ((𝐹(2nd ‘𝐾)𝐺)‘𝑁))
56	55	mptru 1577	. . . 4 ⊢ (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = ((𝐹(2nd ‘𝐾)𝐺)‘𝑁)
57	56	fveq1i 6909	. . 3 ⊢ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅)
58		0lt1o 8540	. . . . . . . . 9 ⊢ ∅ ∈ 1o
59	8	setc1obas 50832	. . . . . . . . 9 ⊢ 1o = (Base‘𝐶)
60	58, 59	eleqtri 2862	. . . . . . . 8 ⊢ ∅ ∈ (Base‘𝐶)
61	60	a1i 11	. . . . . . 7 ⊢ (⊤ → ∅ ∈ (Base‘𝐶))
62	27, 31, 32, 39, 12, 33, 52, 54, 21, 61	postcof2 50705	. . . . . 6 ⊢ (⊤ → (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)))
63	62	mptru 1577	. . . . 5 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅))
64	18	simp3i 1159	. . . . . . . 8 ⊢ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o)
65	64	simpli 489	. . . . . . 7 ⊢ ((1st ‘𝐹)‘∅) = ∅
66	64	simpri 491	. . . . . . 7 ⊢ ((1st ‘𝐺)‘∅) = 1o
67	43, 45	op2ndd 8036	. . . . . . . 8 ⊢ (𝑆 = ⟨( I ↾ 2o), 𝑀⟩ → (2nd ‘𝑆) = 𝑀)
68	34, 67	ax-mp 5	. . . . . . 7 ⊢ (2nd ‘𝑆) = 𝑀
69	65, 66, 68	oveq123i 7457	. . . . . 6 ⊢ (((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅)) = (∅𝑀1o)
70	69, 25	fveq12i 6914	. . . . 5 ⊢ ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)) = ((∅𝑀1o)‘∅)
71	63, 70	eqtri 2786	. . . 4 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((∅𝑀1o)‘∅)
72		0ex 5264	. . . . . . . . 9 ⊢ ∅ ∈ V
73	72	prid1 4723	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
74		df2o3 8512	. . . . . . . 8 ⊢ 2o = {∅, 1o}
75	73, 74	eleqtrri 2863	. . . . . . 7 ⊢ ∅ ∈ 2o
76	75	a1i 11	. . . . . 6 ⊢ (⊤ → ∅ ∈ 2o)
77		1oelpr 8515	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
78	77, 74	eleqtrri 2863	. . . . . . 7 ⊢ 1o ∈ 2o
79	78	a1i 11	. . . . . 6 ⊢ (⊤ → 1o ∈ 2o)
80		eqid 2763	. . . . . . . . . . 11 ⊢ (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, ∅)))
81	80	par2hom 50984	. . . . . . . . . 10 ⊢ ((∅(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, ∅)))∅) = ∅))
82	81	simp3i 1159	. . . . . . . . 9 ⊢ ((∅(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, ∅)))∅) = ∅)
83	82	simpli 489	. . . . . . . 8 ⊢ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o) = 2o
84	75, 83	eleqtrri 2863	. . . . . . 7 ⊢ ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o)
85	84	a1i 11	. . . . . 6 ⊢ (⊤ → ∅ ∈ (∅(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))1o))
86	36, 76, 79, 85	fvmpomptd 50188	. . . . 5 ⊢ (⊤ → ((∅𝑀1o)‘∅) = (𝐴‘∅))
87	86	mptru 1577	. . . 4 ⊢ ((∅𝑀1o)‘∅) = (𝐴‘∅)
88	35	parswapprop 50982	. . . . 5 ⊢ ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)
89	88	simp3i 1159	. . . 4 ⊢ (𝐴‘∅) = 1o
90	71, 87, 89	3eqtri 2790	. . 3 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = 1o
91	57, 90	eqtri 2786	. 2 ⊢ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o
92	26, 91	pm3.2i 476	1 ⊢ (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ∩ cin 4092  ∅c0 4281  ifcif 4485  {csn 4588  {cpr 4590  {ctp 4592  ⟨cop 4838   ↦ cmpt 5189   I cid 5549   × cxp 5653   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024  1_oc1o 8497  2_oc2o 8498  3_oc3o 8499  ndxcnx 17400  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868   Func cfunc 18058  id_funccidfu 18059   ∘_func ccofu 18060   Nat cnat 18148   FuncCat cfuc 18149  SetCatcsetc 18279   ×_c cxpc 18371   curry_F ccurf 18413  Δ_funccdiag 18415   ∘_F cfuco 50643  TermCatctermc 50812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-3o 8506  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-idfu 18063  df-cofu 18064  df-nat 18150  df-fuc 18151  df-setc 18280  df-xpc 18375  df-1stf 18376  df-curf 18417  df-diag 18419  df-fuco 50644  df-thinc 50758  df-termc 50813

This theorem is used by:  fuconuniqvals  50989