Theorem setc1onsubc 49584

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 4359	. . . 4 ⊢ ∅ ⊆ 1o
2		1oex 8421	. . . 4 ⊢ 1o ∈ V
3		setc1onsubc.c	. . . . 5 ⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}
4		df2o3 8419	. . . . 5 ⊢ 2o = {∅, 1o}
5		setc1onsubc.x	. . . . 5 ⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))
6	3, 4, 5	incat 49583	. . . 4 ⊢ ((∅ ⊆ 1o ∧ 1o ∈ V) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)))
7	1, 2, 6	mp2an 692	. . 3 ⊢ (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o))
8	7	simpli 483	. 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 49474	. . . 4 ⊢ 1o = (Base‘𝐸)
13	10, 12	eqtri 2752	. . 3 ⊢ 𝑆 = (Base‘𝐸)
14	9, 13	homffn 17634	. 2 ⊢ 𝐽 Fn (𝑆 × 𝑆)
15		ssid 3966	. . . 4 ⊢ {∅} ⊆ {∅}
16		snsspr1 4774	. . . . . 6 ⊢ {∅} ⊆ {∅, 1o}
17	11	setc1ohomfval 49475	. . . . . . . . 9 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘𝐸)
18		0lt1o 8445	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ 1o)
20	9, 12, 17, 19, 19	homfval 17633	. . . . . . . 8 ⊢ (⊤ → (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅))
21	20	mptru 1547	. . . . . . 7 ⊢ (∅𝐽∅) = (∅{⟨∅, ∅, 1o⟩}∅)
22	2	ovsn2 48842	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
23		df1o2 8418	. . . . . . 7 ⊢ 1o = {∅}
24	21, 22, 23	3eqtri 2756	. . . . . 6 ⊢ (∅𝐽∅) = {∅}
25		setc1onsubc.h	. . . . . . . . 9 ⊢ 𝐻 = (Homf ‘𝐶)
26		snex 5386	. . . . . . . . . 10 ⊢ {∅} ∈ V
27	3, 26	catbas 49208	. . . . . . . . 9 ⊢ {∅} = (Base‘𝐶)
28		snex 5386	. . . . . . . . . 10 ⊢ {⟨∅, ∅, 2o⟩} ∈ V
29	3, 28	cathomfval 49209	. . . . . . . . 9 ⊢ {⟨∅, ∅, 2o⟩} = (Hom ‘𝐶)
30		0ex 5257	. . . . . . . . . . 11 ⊢ ∅ ∈ V
31	30	snid 4622	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
32	31	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ∅ ∈ {∅})
33	25, 27, 29, 32, 32	homfval 17633	. . . . . . . 8 ⊢ (⊤ → (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅))
34	33	mptru 1547	. . . . . . 7 ⊢ (∅𝐻∅) = (∅{⟨∅, ∅, 2o⟩}∅)
35		2oex 8422	. . . . . . . 8 ⊢ 2o ∈ V
36	35	ovsn2 48842	. . . . . . 7 ⊢ (∅{⟨∅, ∅, 2o⟩}∅) = 2o
37	34, 36, 4	3eqtri 2756	. . . . . 6 ⊢ (∅𝐻∅) = {∅, 1o}
38	16, 24, 37	3sstr4i 3995	. . . . 5 ⊢ (∅𝐽∅) ⊆ (∅𝐻∅)
39		oveq1 7376	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐽𝑞) = (∅𝐽𝑞))
40		oveq1 7376	. . . . . . . . 9 ⊢ (𝑝 = ∅ → (𝑝𝐻𝑞) = (∅𝐻𝑞))
41	39, 40	sseq12d 3977	. . . . . . . 8 ⊢ (𝑝 = ∅ → ((𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
42	41	ralbidv 3156	. . . . . . 7 ⊢ (𝑝 = ∅ → (∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞)))
43	30, 42	ralsn 4641	. . . . . 6 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ ∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞))
44		oveq2 7377	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐽𝑞) = (∅𝐽∅))
45		oveq2 7377	. . . . . . . 8 ⊢ (𝑞 = ∅ → (∅𝐻𝑞) = (∅𝐻∅))
46	44, 45	sseq12d 3977	. . . . . . 7 ⊢ (𝑞 = ∅ → ((∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅)))
47	30, 46	ralsn 4641	. . . . . 6 ⊢ (∀𝑞 ∈ {∅} (∅𝐽𝑞) ⊆ (∅𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
48	43, 47	bitri 275	. . . . 5 ⊢ (∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞) ↔ (∅𝐽∅) ⊆ (∅𝐻∅))
49	38, 48	mpbir 231	. . . 4 ⊢ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)
50	23, 12	eqtr3i 2754	. . . . . . . 8 ⊢ {∅} = (Base‘𝐸)
51	9, 50	homffn 17634	. . . . . . 7 ⊢ 𝐽 Fn ({∅} × {∅})
52	51	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 Fn ({∅} × {∅}))
53	25, 27	homffn 17634	. . . . . . 7 ⊢ 𝐻 Fn ({∅} × {∅})
54	53	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐻 Fn ({∅} × {∅}))
55	26	a1i 11	. . . . . 6 ⊢ (⊤ → {∅} ∈ V)
56	52, 54, 55	isssc 17762	. . . . 5 ⊢ (⊤ → (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
57	56	mptru 1547	. . . 4 ⊢ (𝐽 ⊆cat 𝐻 ↔ ({∅} ⊆ {∅} ∧ ∀𝑝 ∈ {∅}∀𝑞 ∈ {∅} (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞)))
58	15, 49, 57	mpbir2an 711	. . 3 ⊢ 𝐽 ⊆cat 𝐻
59		elirr 9526	. . . 4 ⊢ ¬ {∅} ∈ {∅}
60	10, 23	eqtri 2752	. . . . . 6 ⊢ 𝑆 = {∅}
61		biid 261	. . . . . 6 ⊢ (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
62	60, 61	rexeqbii 3315	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
63		rexnal 3082	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
64		fveq2 6840	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = ( 1 ‘∅))
65	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → 1o = {∅})
66		setc1onsubc.i	. . . . . . . . . . . 12 ⊢ 1 = (Id‘𝐶)
67	7	simpri 485	. . . . . . . . . . . 12 ⊢ (Id‘𝐶) = (𝑦 ∈ {∅} ↦ 1o)
68	66, 67	eqtri 2752	. . . . . . . . . . 11 ⊢ 1 = (𝑦 ∈ {∅} ↦ 1o)
69	65, 68, 26	fvmpt 6950	. . . . . . . . . 10 ⊢ (∅ ∈ {∅} → ( 1 ‘∅) = {∅})
70	31, 69	ax-mp 5	. . . . . . . . 9 ⊢ ( 1 ‘∅) = {∅}
71	64, 70	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑥 = ∅ → ( 1 ‘𝑥) = {∅})
72		oveq12 7378	. . . . . . . . . 10 ⊢ ((𝑥 = ∅ ∧ 𝑥 = ∅) → (𝑥𝐽𝑥) = (∅𝐽∅))
73	72	anidms 566	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = (∅𝐽∅))
74	73, 24	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥𝐽𝑥) = {∅})
75	71, 74	eleq12d 2822	. . . . . . 7 ⊢ (𝑥 = ∅ → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ {∅} ∈ {∅}))
76	75	notbid 318	. . . . . 6 ⊢ (𝑥 = ∅ → (¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅}))
77	30, 76	rexsn 4642	. . . . 5 ⊢ (∃𝑥 ∈ {∅} ¬ ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ¬ {∅} ∈ {∅})
78	62, 63, 77	3bitr3ri 302	. . . 4 ⊢ (¬ {∅} ∈ {∅} ↔ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
79	59, 78	mpbi 230	. . 3 ⊢ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)
80		setc1oterm 49473	. . . . . . . 8 ⊢ (SetCat‘1o) ∈ TermCat
81	80	a1i 11	. . . . . . 7 ⊢ (⊤ → (SetCat‘1o) ∈ TermCat)
82	81	termccd 49461	. . . . . 6 ⊢ (⊤ → (SetCat‘1o) ∈ Cat)
83	82	mptru 1547	. . . . 5 ⊢ (SetCat‘1o) ∈ Cat
84	11, 83	eqeltri 2824	. . . 4 ⊢ 𝐸 ∈ Cat
85		setc1onsubc.d	. . . . . 6 ⊢ 𝐷 = (𝐶 ↾cat 𝐽)
86		snex 5386	. . . . . . 7 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} ∈ V
87	3, 86	catcofval 49210	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, · ⟩} = (comp‘𝐶)
88	11	setc1ocofval 49476	. . . . . 6 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘𝐸)
89		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {∅} ↔ 𝑎 = ∅)
90		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
91		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {∅} ↔ 𝑐 = ∅)
92	89, 90, 91	3anbi123i 1155	. . . . . . . . . 10 ⊢ ((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ↔ (𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅))
93	92	anbi1i 624	. . . . . . . . 9 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))))
94		simp1 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑎 = ∅)
95		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑏 = ∅)
96	94, 95	oveq12d 7387	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = (∅𝐽∅))
97	96, 24	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑎𝐽𝑏) = {∅})
98	97	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 ∈ {∅}))
99		velsn 4601	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {∅} ↔ 𝑚 = ∅)
100	98, 99	bitrdi 287	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑚 ∈ (𝑎𝐽𝑏) ↔ 𝑚 = ∅))
101		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → 𝑐 = ∅)
102	95, 101	oveq12d 7387	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = (∅𝐽∅))
103	102, 24	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑏𝐽𝑐) = {∅})
104	103	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 ∈ {∅}))
105		velsn 4601	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {∅} ↔ 𝑛 = ∅)
106	104, 105	bitrdi 287	. . . . . . . . . . 11 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → (𝑛 ∈ (𝑏𝐽𝑐) ↔ 𝑛 = ∅))
107	100, 106	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) → ((𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐)) ↔ (𝑚 = ∅ ∧ 𝑛 = ∅)))
108	107	pm5.32i 574	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
109	93, 108	bitri 275	. . . . . . . 8 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) ↔ ((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)))
110	30	prid1 4722	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1o}
111	110, 4	eleqtrri 2827	. . . . . . . . . . 11 ⊢ ∅ ∈ 2o
112		ineq12 4174	. . . . . . . . . . . . 13 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = (∅ ∩ ∅))
113		0in 4356	. . . . . . . . . . . . 13 ⊢ (∅ ∩ ∅) = ∅
114	112, 113	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝑓 = ∅ ∧ 𝑔 = ∅) → (𝑓 ∩ 𝑔) = ∅)
115	114, 5, 30	ovmpoa 7524	. . . . . . . . . . 11 ⊢ ((∅ ∈ 2o ∧ ∅ ∈ 2o) → (∅ · ∅) = ∅)
116	111, 111, 115	mp2an 692	. . . . . . . . . 10 ⊢ (∅ · ∅) = ∅
117	30	ovsn2 48842	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
118	116, 117	eqtr4i 2755	. . . . . . . . 9 ⊢ (∅ · ∅) = (∅{⟨∅, ∅, ∅⟩}∅)
119		simpl1 1192	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑎 = ∅)
120		simpl2 1193	. . . . . . . . . . . . 13 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑏 = ∅)
121	119, 120	opeq12d 4841	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → ⟨𝑎, 𝑏⟩ = ⟨∅, ∅⟩)
122		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑐 = ∅)
123	121, 122	oveq12d 7387	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅))
124	35, 35	mpoex 8037	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔)) ∈ V
125	5, 124	eqeltri 2824	. . . . . . . . . . . 12 ⊢ · ∈ V
126	125	ovsn2 48842	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}∅) = ·
127	123, 126	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐) = · )
128		simprr 772	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑛 = ∅)
129		simprl 770	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → 𝑚 = ∅)
130	127, 128, 129	oveq123d 7390	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (∅ · ∅))
131	121, 122	oveq12d 7387	. . . . . . . . . . 11 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
132		snex 5386	. . . . . . . . . . . 12 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
133	132	ovsn2 48842	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
134	131, 133	eqtrdi 2780	. . . . . . . . . 10 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐) = {⟨∅, ∅, ∅⟩})
135	134, 128, 129	oveq123d 7390	. . . . . . . . 9 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚) = (∅{⟨∅, ∅, ∅⟩}∅))
136	118, 130, 135	3eqtr4a 2790	. . . . . . . 8 ⊢ (((𝑎 = ∅ ∧ 𝑏 = ∅ ∧ 𝑐 = ∅) ∧ (𝑚 = ∅ ∧ 𝑛 = ∅)) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
137	109, 136	sylbi 217	. . . . . . 7 ⊢ (((𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅}) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
138	137	adantll 714	. . . . . 6 ⊢ (((⊤ ∧ (𝑎 ∈ {∅} ∧ 𝑏 ∈ {∅} ∧ 𝑐 ∈ {∅})) ∧ (𝑚 ∈ (𝑎𝐽𝑏) ∧ 𝑛 ∈ (𝑏𝐽𝑐))) → (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, · ⟩}𝑐)𝑚) = (𝑛(⟨𝑎, 𝑏⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}𝑐)𝑚))
139	84	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐸 ∈ Cat)
140	15	a1i 11	. . . . . 6 ⊢ (⊤ → {∅} ⊆ {∅})
141	85, 27, 50, 9, 87, 88, 138, 139, 140	resccat 49056	. . . . 5 ⊢ (⊤ → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
142	141	mptru 1547	. . . 4 ⊢ (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat)
143	84, 142	mpbir 231	. . 3 ⊢ 𝐷 ∈ Cat
144	58, 79, 143	3pm3.2i 1340	. 2 ⊢ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)
145	8, 14, 144	3pm3.2i 1340	1 ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  {cpr 4587  {ctp 4589  ⟨cop 4591  ⟨cotp 4593   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1_oc1o 8404  2_oc2o 8405  ndxcnx 17139  Basecbs 17155  Hom chom 17207  compcco 17208  Catccat 17605  Idccid 17606  Hom_f chomf 17607   ⊆_cat cssc 17749   ↾_cat cresc 17750  SetCatcsetc 18017  TermCatctermc 49454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-hom 17220  df-cco 17221  df-cat 17609  df-cid 17610  df-homf 17611  df-comf 17612  df-ssc 17752  df-resc 17753  df-setc 18018  df-thinc 49400  df-termc 49455

This theorem is referenced by:  cnelsubc  49586