Theorem fuconuniqvalslem 50874

Description: Lemma for fuconuniqvals 50875. (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 2769	. . . . . . . 8 ⊢ (⟨𝐶, 𝐶⟩ ∘F 𝐸) = 𝐿
3	2	fveq2i 6888	. . . . . . 7 ⊢ (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿)
4	3	a1i 11	. . . . . 6 ⊢ (⊤ → (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = (2nd ‘𝐿))
5		eqid 2760	. . . . . 6 ⊢ (𝐶 FuncCat 𝐶) = (𝐶 FuncCat 𝐶)
6		eqid 2760	. . . . . 6 ⊢ (Id‘(𝐶 FuncCat 𝐶)) = (Id‘(𝐶 FuncCat 𝐶))
7		eqid 2760	. . . . . 6 ⊢ (idfunc‘𝐶) = (idfunc‘𝐶)
8		fuconuniq2.c	. . . . . . . . 9 ⊢ 𝐶 = (SetCat‘1o)
9		setc1oterm 50717	. . . . . . . . 9 ⊢ (SetCat‘1o) ∈ TermCat
10	8, 9	eqeltri 2856	. . . . . . . 8 ⊢ 𝐶 ∈ TermCat
11	10	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐶 ∈ TermCat)
12	11	termccatd 50705	. . . . . 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 50871	. . . . . . . . 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 50577	. . . . 5 ⊢ (⊤ → (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = 𝑁)
23	22	mptru 1576	. . . 4 ⊢ (𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶))) = 𝑁
24	23	fveq1i 6886	. . 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 2783	. 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 50858	. . . . . . . 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 50869	. . . . . . . . . 10 ⊢ ⟨( I ↾ 2o), 𝑀⟩ ∈ (𝐸 Func 𝐸)
38	34, 37	eqeltri 2856	. . . . . . . . 9 ⊢ 𝑆 ∈ (𝐸 Func 𝐸)
39	38	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝑆 ∈ (𝐸 Func 𝐸))
40		mptresid 6042	. . . . . . . . . . 11 ⊢ ( I ↾ 2o) = (𝑥 ∈ 2o ↦ 𝑥)
41		2oex 8490	. . . . . . . . . . . 12 ⊢ 2o ∈ V
42	41	mptex 7229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 2o ↦ 𝑥) ∈ V
43	40, 42	eqeltri 2856	. . . . . . . . . 10 ⊢ ( I ↾ 2o) ∈ V
44	41, 41	mpoex 8092	. . . . . . . . . . 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 2856	. . . . . . . . . 10 ⊢ 𝑀 ∈ V
46	43, 45	op1std 8011	. . . . . . . . 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 50763	. . . . . . 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 50584	. . . . 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 1576	. . . 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 6886	. . 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 8514	. . . . . . . . 9 ⊢ ∅ ∈ 1o
59	8	setc1obas 50718	. . . . . . . . 9 ⊢ 1o = (Base‘𝐶)
60	58, 59	eleqtri 2858	. . . . . . . 8 ⊢ ∅ ∈ (Base‘𝐶)
61	60	a1i 11	. . . . . . 7 ⊢ (⊤ → ∅ ∈ (Base‘𝐶))
62	27, 31, 32, 39, 12, 33, 52, 54, 21, 61	postcof2 50591	. . . . . 6 ⊢ (⊤ → (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)))
63	62	mptru 1576	. . . . 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 8012	. . . . . . . 8 ⊢ (𝑆 = ⟨( I ↾ 2o), 𝑀⟩ → (2nd ‘𝑆) = 𝑀)
68	34, 67	ax-mp 5	. . . . . . 7 ⊢ (2nd ‘𝑆) = 𝑀
69	65, 66, 68	oveq123i 7434	. . . . . 6 ⊢ (((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅)) = (∅𝑀1o)
70	69, 25	fveq12i 6891	. . . . 5 ⊢ ((((1st ‘𝐹)‘∅)(2nd ‘𝑆)((1st ‘𝐺)‘∅))‘(𝑁‘∅)) = ((∅𝑀1o)‘∅)
71	63, 70	eqtri 2783	. . . 4 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = ((∅𝑀1o)‘∅)
72		0ex 5260	. . . . . . . . 9 ⊢ ∅ ∈ V
73	72	prid1 4722	. . . . . . . 8 ⊢ ∅ ∈ {∅, 1o}
74		df2o3 8486	. . . . . . . 8 ⊢ 2o = {∅, 1o}
75	73, 74	eleqtrri 2859	. . . . . . 7 ⊢ ∅ ∈ 2o
76	75	a1i 11	. . . . . 6 ⊢ (⊤ → ∅ ∈ 2o)
77		1oelpr 8489	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
78	77, 74	eleqtrri 2859	. . . . . . 7 ⊢ 1o ∈ 2o
79	78	a1i 11	. . . . . 6 ⊢ (⊤ → 1o ∈ 2o)
80		eqid 2760	. . . . . . . . . . 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 50870	. . . . . . . . . 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 2859	. . . . . . 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 50074	. . . . 5 ⊢ (⊤ → ((∅𝑀1o)‘∅) = (𝐴‘∅))
87	86	mptru 1576	. . . 4 ⊢ ((∅𝑀1o)‘∅) = (𝐴‘∅)
88	35	parswapprop 50868	. . . . 5 ⊢ ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)
89	88	simp3i 1159	. . . 4 ⊢ (𝐴‘∅) = 1o
90	71, 87, 89	3eqtri 2787	. . 3 ⊢ (((𝐹(2nd ‘𝐾)𝐺)‘𝑁)‘∅) = 1o
91	57, 90	eqtri 2783	. 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 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∩ cin 4089  ∅c0 4278  ifcif 4481  {csn 4583  {cpr 4585  {ctp 4587  ⟨cop 4589   ↦ 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  id_funccidfu 18032   ∘_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:  fuconuniqvals  50875