Theorem setc1onsubc 50439

Description: Construct a category with one object and two morphisms and prove that category (SetCat‘1_o) satisfies all conditions for a subcategory but the compatibility of identity morphisms, showing the necessity of the latter condition in defining a subcategory. Exercise 4A of [Adamek] p. 58. (Contributed by Zhi Wang, 6-Nov-2025.)

Hypotheses

Ref	Expression
setc1onsubc.c	⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}
setc1onsubc.x	⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))
setc1onsubc.e	⊢ 𝐸 = (SetCat‘1o)
setc1onsubc.j	⊢ 𝐽 = (Homf ‘𝐸)
setc1onsubc.s	⊢ 𝑆 = 1o
setc1onsubc.h	⊢ 𝐻 = (Homf ‘𝐶)
setc1onsubc.i	⊢ 1 = (Id‘𝐶)
setc1onsubc.d	⊢ 𝐷 = (𝐶 ↾cat 𝐽)

Assertion

Ref	Expression
setc1onsubc	⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))

Distinct variable group:   𝑓,𝑔

Allowed substitution hints:   𝐶(𝑥, 𝑓, 𝑔)   𝐷(𝑥, 𝑓, 𝑔)   𝑆(𝑥, 𝑓, 𝑔)   · (𝑥, 𝑓, 𝑔)   1 (𝑥, 𝑓, 𝑔)   𝐸(𝑥, 𝑓, 𝑔)   𝐻(𝑥, 𝑓, 𝑔)   𝐽(𝑥, 𝑓, 𝑔)

Proof of Theorem setc1onsubc

Dummy variables 𝑦 𝑎 𝑏 𝑐 𝑚 𝑛 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4357	. . . 4 ⊢ ∅ ⊆ 1o
2		1oex 8469	. . . 4 ⊢ 1o ∈ V
3		setc1onsubc.c	. . . . 5 ⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}
4		df2o3 8467	. . . . 5 ⊢ 2o = {∅, 1o}
5		setc1onsubc.x	. . . . 5 ⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))
6	3, 4, 5	incat 50438	. . . 4 ⊢ ((∅ ⊆ 1o ∧ 1o ∈ V) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)))
7	1, 2, 6	mp2an 705	. . 3 ⊢ (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o))
8	7	simpli 489	. 2 ⊢ 𝐶 ∈ Cat
9		setc1onsubc.j	. . 3 ⊢ 𝐽 = (Homf ‘𝐸)
10		setc1onsubc.s	. . . 4 ⊢ 𝑆 = 1o
11		setc1onsubc.e	. . . . 5 ⊢ 𝐸 = (SetCat‘1o)
12	11	setc1obas 50329	. . . 4 ⊢ 1o = (Base‘𝐸)
13	10, 12	eqtri 2788	. . 3 ⊢ 𝑆 = (Base‘𝐸)
14	9, 13	homffn 17773	. 2 ⊢ 𝐽 Fn (𝑆 × 𝑆)
15		ssid 3960	. . . 4 ⊢ {∅} ⊆ {∅}
16		snsspr1 4782	. . . . . 6 ⊢ {∅} ⊆ {∅, 1o}
17	11	setc1ohomfval 50330	. . . . . . . . 9 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘𝐸)
18		0lt1o 8495	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ 1o)
20	9, 12, 17, 19, 19	homfval 17772	. . . . . . . 8 ⊢ (⊤ → (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅))
21	20	mptru 1577	. . . . . . 7 ⊢ (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅)
22	2	ovsn2 49698	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
23		df1o2 8466	. . . . . . 7 ⊢ 1o = {∅}
24	21, 22, 23	3eqtri 2792	. . . . . 6 ⊢ (∅𝐽∅) = {∅}
25		setc1onsubc.h	. . . . . . . . 9 ⊢ 𝐻 = (Homf ‘𝐶)
26		snex 5412	. . . . . . . . . 10 ⊢ {∅} ∈ V
27	3, 26	catbas 50063	. . . . . . . . 9 ⊢ {∅} = (Base‘𝐶)
28		snex 5412	. . . . . . . . . 10 ⊢ {⟨∅, ∅, 2o⟩} ∈ V
29	3, 28	cathomfval 50064	. . . . . . . . 9 ⊢ {⟨∅, ∅, 2o⟩} = (Hom ‘𝐶)
30		0ex 5272	. . . . . . . . . . 11 ⊢ ∅ ∈ V
31	30	snid 4630	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
32	31	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ {∅})
33	25, 27, 29, 32, 32	homfval 17772	. . . . . . . 8 ⊢ (⊤ → (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅))
34	33	mptru 1577	. . . . . . 7 ⊢ (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅)
35		2oex 8471	. . . . . . . 8 ⊢ 2o ∈ V
36	35	ovsn2 49698	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 2o⟩}∅) = 2o
37	34, 36, 4	3eqtri 2792	. . . . . 6 ⊢ (∅𝐻∅) = {∅, 1o}
38	16, 24, 37	3sstr4i 3989	. . . . 5 ⊢ (∅𝐽∅) ⊆ (∅𝐻∅)
39		oveq1 7426	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐽𝑞) = (∅𝐽𝑞))
40		oveq1 7426	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐻𝑞) = (∅𝐻𝑞))
41	39, 40	sseq12d 3971	. . . . . . . 8 ⊢ (𝑝 = ∅ → ((𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
42	41	ralbidv 3190	. . . . . . 7 ⊢ (𝑝 = ∅ → (∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
43	30, 42	ralsn 4649	. . . . . 6 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞))
44		oveq2 7427	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐽𝑞) = (∅𝐽∅))
45		oveq2 7427	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐻𝑞) = (∅𝐻∅))
46	44, 45	sseq12d 3971	. . . . . . 7 ⊢ (𝑞 = ∅ → ((∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅)))
47	30, 46	ralsn 4649	. . . . . 6 ⊢ (∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
48	43, 47	bitri 278	. . . . 5 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
49	38, 48	mpbir 234	. . . 4 ⊢ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)
50	23, 12	eqtr3i 2790	. . . . . . . 8 ⊢ {∅} = (Base‘𝐸)
51	9, 50	homffn 17773	. . . . . . 7 ⊢ 𝐽 Fn ({∅} × {∅})
52	51	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 Fn ({∅} × {∅}))
53	25, 27	homffn 17773	. . . . . . 7 ⊢ 𝐻 Fn ({∅} × {∅})
54	53	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐻 Fn ({∅} × {∅}))
55	26	a1i 11	. . . . . 6 ⊢ (⊤ → {∅} ∈ V)
56	52, 54, 55	isssc 17901	. . . . 5 ⊢ (⊤ → (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
57	56	mptru 1577	. . . 4 ⊢ (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)))
58	15, 49, 57	mpbir2an 724	. . 3 ⊢ 𝐽 ⊆cat 𝐻
59		1on 8472	. . . . . 6 ⊢ 1o ∈ On
60	23, 59	eqeltrri 2862	. . . . 5 ⊢ {∅} ∈ On
61	60	onirri 6479	. . . 4 ⊢ ¬ {∅} ∈ {∅}
62	10, 23	eqtri 2788	. . . . . 6 ⊢ 𝑆 = {∅}
63		biid 264	. . . . . 6 ⊢ (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
64	62, 63	rexeqbii 3339	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
65		rexnal 3119	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
66		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = ( 1 ‘∅))
67	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → 1o = {∅})
68		setc1onsubc.i	. . . . . . . . . . . 12 ⊢ 1 = (Id‘𝐶)
69	7	simpri 491	. . . . . . . . . . . 12 ⊢ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)
70	68, 69	eqtri 2788	. . . . . . . . . . 11 ⊢ 1 = (𝑦 ∈ {∅} ↦ 1o)
71	67, 70, 26	fvmpt 6993	. . . . . . . . . 10 ⊢ (∅ ∈ {∅} → ( 1 ‘∅) = {∅})
72	31, 71	ax-mp 5	. . . . . . . . 9 ⊢ ( 1 ‘∅) = {∅}
73	66, 72	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = {∅})
74		oveq12 7428	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑥 = ∅) → (𝑥𝐽𝑥) = (∅𝐽∅))
75	74	anidms 577	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = (∅𝐽∅))
76	75, 24	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = {∅})
77	73, 76	eleq12d 2859	. . . . . . 7 ⊢ (𝑥 = ∅ → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ {∅} ∈ {∅}))
78	77	notbid 321	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅}))
79	30, 78	rexsn 4650	. . . . 5 ⊢ (∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅})
80	64, 65, 79	3bitr3ri 305	. . . 4 ⊢ (¬ {∅} ∈ {∅} ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
81	61, 80	mpbi 233	. . 3 ⊢ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)
82		setc1oterm 50328	. . . . . . . 8 ⊢ (SetCat‘1o) ∈ TermCat
83	82	a1i 11	. . . . . . 7 ⊢ (⊤ → (SetCat‘1o) ∈ TermCat)
84	83	termccd 50316	. . . . . 6 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
85	84	mptru 1577	. . . . 5 ⊢ (SetCat‘1o) ∈ Cat
86	11, 85	eqeltri 2861	. . . 4 ⊢ 𝐸 ∈ Cat
87		setc1onsubc.d	. . . . . 6 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
88		snex 5412	. . . . . . 7 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} ∈ V
89	3, 88	catcofval 50065	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} = (comp‘𝐶)
90	11	setc1ocofval 50331	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘𝐸)
91		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {∅} ↔ 𝑎 = ∅)
92		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
93		velsn 4607	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {∅} ↔ 𝑐 = ∅)
94	91, 92, 93	3anbi123i 1173	. . . . . . . . . 10 ⊢ ((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ↔ (𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅))
95	94	anbi1i 636	. . . . . . . . 9 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))))
96		simp1 1154	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑎 = ∅)
97		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑏 = ∅)
98	96, 97	oveq12d 7437	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = (∅𝐽∅))
99	98, 24	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = {∅})
100	99	eleq2d 2851	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 ∈ {∅}))
101		velsn 4607	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {∅} ↔ 𝑚 = ∅)
102	100, 101	bitrdi 290	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 = ∅))
103		simp3 1156	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑐 = ∅)
104	97, 103	oveq12d 7437	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = (∅𝐽∅))
105	104, 24	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = {∅})
106	105	eleq2d 2851	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 ∈ {∅}))
107		velsn 4607	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {∅} ↔ 𝑛 = ∅)
108	106, 107	bitrdi 290	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 = ∅))
109	102, 108	anbi12d 644	. . . . . . . . . 10 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → ((𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐)) ↔ (𝑚 = ∅ ∧ 𝑛 = ∅)))
110	109	pm5.32i 585	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
111	95, 110	bitri 278	. . . . . . . 8 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
112	30	prid1 4730	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
113	112, 4	eleqtrri 2864	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
114		ineq12 4168	. . . . . . . . . . . . 13 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = (∅ ∩ ∅))
115		0in 4354	. . . . . . . . . . . . 13 ⊢ (∅ ∩ ∅) = ∅
116	114, 115	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = ∅)
117	116, 5, 30	ovmpoa 7574	. . . . . . . . . . 11 ⊢ ((∅ ∈ 2o ∧ ∅ ∈ 2o) → (∅ · ∅) = ∅)
118	113, 113, 117	mp2an 705	. . . . . . . . . 10 ⊢ (∅ · ∅) = ∅
119	30	ovsn2 49698	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
120	118, 119	eqtr4i 2791	. . . . . . . . 9 ⊢ (∅ · ∅) = (∅{⟨∅, ∅, ∅⟩}∅)
121		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑎 = ∅)
122		simpl2 1211	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑏 = ∅)
123	121, 122	opeq12d 4848	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → ⟨𝑎, 𝑏⟩ = ⟨∅, ∅⟩)
124		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑐 = ∅)
125	123, 124	oveq12d 7437	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅))
126	35, 35	mpoex 8082	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔)) ∈ V
127	5, 126	eqeltri 2861	. . . . . . . . . . . 12 ⊢ · ∈ V
128	127	ovsn2 49698	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅) = ·
129	125, 128	eqtrdi 2816	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = · )
130		simprr 785	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑛 = ∅)
131		simprl 783	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑚 = ∅)
132	129, 130, 131	oveq123d 7440	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (∅ · ∅))
133	123, 124	oveq12d 7437	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
134		snex 5412	. . . . . . . . . . . 12 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
135	134	ovsn2 49698	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
136	133, 135	eqtrdi 2816	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = {⟨∅, ∅, ∅⟩})
137	136, 130, 131	oveq123d 7440	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚) = (∅{⟨∅, ∅, ∅⟩}∅))
138	120, 132, 137	3eqtr4a 2826	. . . . . . . 8 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
139	111, 138	sylbi 220	. . . . . . 7 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
140	139	adantll 727	. . . . . 6 ⊢ (((⊤ ∧ (𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅})) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
141	86	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐸 ∈ Cat)
142	15	a1i 11	. . . . . 6 ⊢ (⊤ → {∅} ⊆ {∅})
143	87, 27, 50, 9, 89, 90, 140, 141, 142	resccat 49911	. . . . 5 ⊢ (⊤ → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
144	143	mptru 1577	. . . 4 ⊢ (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat)
145	86, 144	mpbir 234	. . 3 ⊢ 𝐷 ∈ Cat
146	58, 81, 145	3pm3.2i 1358	. 2 ⊢ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)
147	8, 14, 146	3pm3.2i 1358	1 ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  {cpr 4593  {ctp 4595  ⟨cop 4597  ⟨cotp 4599   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  Oncon0 6364   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1_oc1o 8452  2_oc2o 8453  ndxcnx 17277  Basecbs 17293  Hom chom 17345  compcco 17346  Catccat 17744  Idccid 17745  Hom_f chomf 17746   ⊆_cat cssc 17888   ↾_cat cresc 17889  SetCatcsetc 18156  TermCatctermc 50309

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-hom 17358  df-cco 17359  df-cat 17748  df-cid 17749  df-homf 17750  df-comf 17751  df-ssc 17891  df-resc 17892  df-setc 18157  df-thinc 50255  df-termc 50310

This theorem is used by:  cnelsubc  50441