Theorem setc1onsubc 50100

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 4329	. . . 4 ⊢ ∅ ⊆ 1o
2		1oex 8406	. . . 4 ⊢ 1o ∈ V
3		setc1onsubc.c	. . . . 5 ⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}
4		df2o3 8404	. . . . 5 ⊢ 2o = {∅, 1o}
5		setc1onsubc.x	. . . . 5 ⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))
6	3, 4, 5	incat 50099	. . . 4 ⊢ ((∅ ⊆ 1o ∧ 1o ∈ V) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)))
7	1, 2, 6	mp2an 698	. . 3 ⊢ (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o))
8	7	simpli 484	. 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 49990	. . . 4 ⊢ 1o = (Base‘𝐸)
13	10, 12	eqtri 2762	. . 3 ⊢ 𝑆 = (Base‘𝐸)
14	9, 13	homffn 17651	. 2 ⊢ 𝐽 Fn (𝑆 × 𝑆)
15		ssid 3937	. . . 4 ⊢ {∅} ⊆ {∅}
16		snsspr1 4746	. . . . . 6 ⊢ {∅} ⊆ {∅, 1o}
17	11	setc1ohomfval 49991	. . . . . . . . 9 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘𝐸)
18		0lt1o 8430	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ 1o)
20	9, 12, 17, 19, 19	homfval 17650	. . . . . . . 8 ⊢ (⊤ → (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅))
21	20	mptru 1554	. . . . . . 7 ⊢ (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅)
22	2	ovsn2 49359	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
23		df1o2 8403	. . . . . . 7 ⊢ 1o = {∅}
24	21, 22, 23	3eqtri 2766	. . . . . 6 ⊢ (∅𝐽∅) = {∅}
25		setc1onsubc.h	. . . . . . . . 9 ⊢ 𝐻 = (Homf ‘𝐶)
26		snex 5369	. . . . . . . . . 10 ⊢ {∅} ∈ V
27	3, 26	catbas 49724	. . . . . . . . 9 ⊢ {∅} = (Base‘𝐶)
28		snex 5369	. . . . . . . . . 10 ⊢ {⟨∅, ∅, 2o⟩} ∈ V
29	3, 28	cathomfval 49725	. . . . . . . . 9 ⊢ {⟨∅, ∅, 2o⟩} = (Hom ‘𝐶)
30		0ex 5230	. . . . . . . . . . 11 ⊢ ∅ ∈ V
31	30	snid 4595	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
32	31	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ {∅})
33	25, 27, 29, 32, 32	homfval 17650	. . . . . . . 8 ⊢ (⊤ → (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅))
34	33	mptru 1554	. . . . . . 7 ⊢ (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅)
35		2oex 8407	. . . . . . . 8 ⊢ 2o ∈ V
36	35	ovsn2 49359	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 2o⟩}∅) = 2o
37	34, 36, 4	3eqtri 2766	. . . . . 6 ⊢ (∅𝐻∅) = {∅, 1o}
38	16, 24, 37	3sstr4i 3966	. . . . 5 ⊢ (∅𝐽∅) ⊆ (∅𝐻∅)
39		oveq1 7364	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐽𝑞) = (∅𝐽𝑞))
40		oveq1 7364	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐻𝑞) = (∅𝐻𝑞))
41	39, 40	sseq12d 3948	. . . . . . . 8 ⊢ (𝑝 = ∅ → ((𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
42	41	ralbidv 3162	. . . . . . 7 ⊢ (𝑝 = ∅ → (∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
43	30, 42	ralsn 4614	. . . . . 6 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞))
44		oveq2 7365	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐽𝑞) = (∅𝐽∅))
45		oveq2 7365	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐻𝑞) = (∅𝐻∅))
46	44, 45	sseq12d 3948	. . . . . . 7 ⊢ (𝑞 = ∅ → ((∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅)))
47	30, 46	ralsn 4614	. . . . . 6 ⊢ (∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
48	43, 47	bitri 276	. . . . 5 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
49	38, 48	mpbir 232	. . . 4 ⊢ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)
50	23, 12	eqtr3i 2764	. . . . . . . 8 ⊢ {∅} = (Base‘𝐸)
51	9, 50	homffn 17651	. . . . . . 7 ⊢ 𝐽 Fn ({∅} × {∅})
52	51	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 Fn ({∅} × {∅}))
53	25, 27	homffn 17651	. . . . . . 7 ⊢ 𝐻 Fn ({∅} × {∅})
54	53	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐻 Fn ({∅} × {∅}))
55	26	a1i 11	. . . . . 6 ⊢ (⊤ → {∅} ∈ V)
56	52, 54, 55	isssc 17779	. . . . 5 ⊢ (⊤ → (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
57	56	mptru 1554	. . . 4 ⊢ (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)))
58	15, 49, 57	mpbir2an 717	. . 3 ⊢ 𝐽 ⊆cat 𝐻
59		1on 8408	. . . . . 6 ⊢ 1o ∈ On
60	23, 59	eqeltrri 2836	. . . . 5 ⊢ {∅} ∈ On
61	60	onirri 6425	. . . 4 ⊢ ¬ {∅} ∈ {∅}
62	10, 23	eqtri 2762	. . . . . 6 ⊢ 𝑆 = {∅}
63		biid 262	. . . . . 6 ⊢ (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
64	62, 63	rexeqbii 3312	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
65		rexnal 3091	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
66		fveq2 6828	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = ( 1 ‘∅))
67	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → 1o = {∅})
68		setc1onsubc.i	. . . . . . . . . . . 12 ⊢ 1 = (Id‘𝐶)
69	7	simpri 486	. . . . . . . . . . . 12 ⊢ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)
70	68, 69	eqtri 2762	. . . . . . . . . . 11 ⊢ 1 = (𝑦 ∈ {∅} ↦ 1o)
71	67, 70, 26	fvmpt 6936	. . . . . . . . . 10 ⊢ (∅ ∈ {∅} → ( 1 ‘∅) = {∅})
72	31, 71	ax-mp 5	. . . . . . . . 9 ⊢ ( 1 ‘∅) = {∅}
73	66, 72	eqtrdi 2790	. . . . . . . 8 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = {∅})
74		oveq12 7366	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑥 = ∅) → (𝑥𝐽𝑥) = (∅𝐽∅))
75	74	anidms 571	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = (∅𝐽∅))
76	75, 24	eqtrdi 2790	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = {∅})
77	73, 76	eleq12d 2833	. . . . . . 7 ⊢ (𝑥 = ∅ → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ {∅} ∈ {∅}))
78	77	notbid 319	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅}))
79	30, 78	rexsn 4615	. . . . 5 ⊢ (∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅})
80	64, 65, 79	3bitr3ri 303	. . . 4 ⊢ (¬ {∅} ∈ {∅} ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
81	61, 80	mpbi 231	. . 3 ⊢ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)
82		setc1oterm 49989	. . . . . . . 8 ⊢ (SetCat‘1o) ∈ TermCat
83	82	a1i 11	. . . . . . 7 ⊢ (⊤ → (SetCat‘1o) ∈ TermCat)
84	83	termccd 49977	. . . . . 6 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
85	84	mptru 1554	. . . . 5 ⊢ (SetCat‘1o) ∈ Cat
86	11, 85	eqeltri 2835	. . . 4 ⊢ 𝐸 ∈ Cat
87		setc1onsubc.d	. . . . . 6 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
88		snex 5369	. . . . . . 7 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} ∈ V
89	3, 88	catcofval 49726	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} = (comp‘𝐶)
90	11	setc1ocofval 49992	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘𝐸)
91		velsn 4572	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {∅} ↔ 𝑎 = ∅)
92		velsn 4572	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
93		velsn 4572	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {∅} ↔ 𝑐 = ∅)
94	91, 92, 93	3anbi123i 1161	. . . . . . . . . 10 ⊢ ((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ↔ (𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅))
95	94	anbi1i 630	. . . . . . . . 9 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))))
96		simp1 1142	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑎 = ∅)
97		simp2 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑏 = ∅)
98	96, 97	oveq12d 7375	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = (∅𝐽∅))
99	98, 24	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = {∅})
100	99	eleq2d 2825	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 ∈ {∅}))
101		velsn 4572	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {∅} ↔ 𝑚 = ∅)
102	100, 101	bitrdi 288	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 = ∅))
103		simp3 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑐 = ∅)
104	97, 103	oveq12d 7375	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = (∅𝐽∅))
105	104, 24	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = {∅})
106	105	eleq2d 2825	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 ∈ {∅}))
107		velsn 4572	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {∅} ↔ 𝑛 = ∅)
108	106, 107	bitrdi 288	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 = ∅))
109	102, 108	anbi12d 638	. . . . . . . . . 10 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → ((𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐)) ↔ (𝑚 = ∅ ∧ 𝑛 = ∅)))
110	109	pm5.32i 579	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
111	95, 110	bitri 276	. . . . . . . 8 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
112	30	prid1 4695	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
113	112, 4	eleqtrri 2838	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
114		ineq12 4145	. . . . . . . . . . . . 13 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = (∅ ∩ ∅))
115		0in 4326	. . . . . . . . . . . . 13 ⊢ (∅ ∩ ∅) = ∅
116	114, 115	eqtrdi 2790	. . . . . . . . . . . 12 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = ∅)
117	116, 5, 30	ovmpoa 7512	. . . . . . . . . . 11 ⊢ ((∅ ∈ 2o ∧ ∅ ∈ 2o) → (∅ · ∅) = ∅)
118	113, 113, 117	mp2an 698	. . . . . . . . . 10 ⊢ (∅ · ∅) = ∅
119	30	ovsn2 49359	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
120	118, 119	eqtr4i 2765	. . . . . . . . 9 ⊢ (∅ · ∅) = (∅{⟨∅, ∅, ∅⟩}∅)
121		simpl1 1198	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑎 = ∅)
122		simpl2 1199	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑏 = ∅)
123	121, 122	opeq12d 4813	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → ⟨𝑎, 𝑏⟩ = ⟨∅, ∅⟩)
124		simpl3 1200	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑐 = ∅)
125	123, 124	oveq12d 7375	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅))
126	35, 35	mpoex 8022	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔)) ∈ V
127	5, 126	eqeltri 2835	. . . . . . . . . . . 12 ⊢ · ∈ V
128	127	ovsn2 49359	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅) = ·
129	125, 128	eqtrdi 2790	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = · )
130		simprr 778	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑛 = ∅)
131		simprl 776	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑚 = ∅)
132	129, 130, 131	oveq123d 7378	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (∅ · ∅))
133	123, 124	oveq12d 7375	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
134		snex 5369	. . . . . . . . . . . 12 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
135	134	ovsn2 49359	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
136	133, 135	eqtrdi 2790	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = {⟨∅, ∅, ∅⟩})
137	136, 130, 131	oveq123d 7378	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚) = (∅{⟨∅, ∅, ∅⟩}∅))
138	120, 132, 137	3eqtr4a 2800	. . . . . . . 8 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
139	111, 138	sylbi 218	. . . . . . 7 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
140	139	adantll 720	. . . . . 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 49572	. . . . 5 ⊢ (⊤ → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
144	143	mptru 1554	. . . 4 ⊢ (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat)
145	86, 144	mpbir 232	. . 3 ⊢ 𝐷 ∈ Cat
146	58, 81, 145	3pm3.2i 1346	. 2 ⊢ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)
147	8, 14, 146	3pm3.2i 1346	1 ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  ∅c0 4262  {csn 4556  {cpr 4558  {ctp 4560  ⟨cop 4562  ⟨cotp 4564   class class class wbr 5073   ↦ cmpt 5154   × cxp 5617  Oncon0 6311   Fn wfn 6481  ‘cfv 6486  (class class class)co 7357   ∈ cmpo 7359  1_oc1o 8389  2_oc2o 8390  ndxcnx 17155  Basecbs 17171  Hom chom 17223  compcco 17224  Catccat 17622  Idccid 17623  Hom_f chomf 17624   ⊆_cat cssc 17766   ↾_cat cresc 17767  SetCatcsetc 18034  TermCatctermc 49970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-ot 4565  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-hom 17236  df-cco 17237  df-cat 17626  df-cid 17627  df-homf 17628  df-comf 17629  df-ssc 17769  df-resc 17770  df-setc 18035  df-thinc 49916  df-termc 49971

This theorem is referenced by:  cnelsubc  50102