Theorem ust0 24228

Description: The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ust0	⊢ (UnifOn‘∅) = {{∅}}

Proof of Theorem ust0

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5307	. . . . . . . 8 ⊢ ∅ ∈ V
2		isust 24212	. . . . . . . 8 ⊢ (∅ ∈ V → (𝑢 ∈ (UnifOn‘∅) ↔ (𝑢 ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (𝑢 ∈ (UnifOn‘∅) ↔ (𝑢 ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
4	3	simp1bi 1146	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ 𝒫 (∅ × ∅))
5		0xp 5784	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
6	5	pweqi 4616	. . . . . . 7 ⊢ 𝒫 (∅ × ∅) = 𝒫 ∅
7		pw0 4812	. . . . . . 7 ⊢ 𝒫 ∅ = {∅}
8	6, 7	eqtri 2765	. . . . . 6 ⊢ 𝒫 (∅ × ∅) = {∅}
9	4, 8	sseqtrdi 4024	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ {∅})
10		ustbasel 24215	. . . . . . 7 ⊢ (𝑢 ∈ (UnifOn‘∅) → (∅ × ∅) ∈ 𝑢)
11	5, 10	eqeltrrid 2846	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → ∅ ∈ 𝑢)
12	11	snssd 4809	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → {∅} ⊆ 𝑢)
13	9, 12	eqssd 4001	. . . 4 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 = {∅})
14		velsn 4642	. . . 4 ⊢ (𝑢 ∈ {{∅}} ↔ 𝑢 = {∅})
15	13, 14	sylibr 234	. . 3 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ∈ {{∅}})
16	15	ssriv 3987	. 2 ⊢ (UnifOn‘∅) ⊆ {{∅}}
17	8	eqimss2i 4045	. . . 4 ⊢ {∅} ⊆ 𝒫 (∅ × ∅)
18	1	snid 4662	. . . . 5 ⊢ ∅ ∈ {∅}
19	5, 18	eqeltri 2837	. . . 4 ⊢ (∅ × ∅) ∈ {∅}
20	18	a1i 11	. . . . . 6 ⊢ (∅ ⊆ ∅ → ∅ ∈ {∅})
21	8	raleqi 3324	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ {∅} (∅ ⊆ 𝑤 → 𝑤 ∈ {∅}))
22		sseq2 4010	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (∅ ⊆ 𝑤 ↔ ∅ ⊆ ∅))
23		eleq1 2829	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
24	22, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅})))
25	1, 24	ralsn 4681	. . . . . . 7 ⊢ (∀𝑤 ∈ {∅} (∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
26	21, 25	bitri 275	. . . . . 6 ⊢ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
27	20, 26	mpbir 231	. . . . 5 ⊢ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅})
28		inidm 4227	. . . . . . 7 ⊢ (∅ ∩ ∅) = ∅
29	28, 18	eqeltri 2837	. . . . . 6 ⊢ (∅ ∩ ∅) ∈ {∅}
30		ineq2 4214	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∅ ∩ 𝑤) = (∅ ∩ ∅))
31	30	eleq1d 2826	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅}))
32	1, 31	ralsn 4681	. . . . . 6 ⊢ (∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅})
33	29, 32	mpbir 231	. . . . 5 ⊢ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}
34		res0 6001	. . . . . . 7 ⊢ ( I ↾ ∅) = ∅
35	34	eqimssi 4044	. . . . . 6 ⊢ ( I ↾ ∅) ⊆ ∅
36		cnv0 6160	. . . . . . 7 ⊢ ◡∅ = ∅
37	36, 18	eqeltri 2837	. . . . . 6 ⊢ ◡∅ ∈ {∅}
38		0trrel 15020	. . . . . . 7 ⊢ (∅ ∘ ∅) ⊆ ∅
39		id 22	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → 𝑤 = ∅)
40	39, 39	coeq12d 5875	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∘ 𝑤) = (∅ ∘ ∅))
41	40	sseq1d 4015	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((𝑤 ∘ 𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅))
42	1, 41	rexsn 4682	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅)
43	38, 42	mpbir 231	. . . . . 6 ⊢ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅
44	35, 37, 43	3pm3.2i 1340	. . . . 5 ⊢ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅)
45		sseq1 4009	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ⊆ 𝑤 ↔ ∅ ⊆ 𝑤))
46	45	imbi1d 341	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
47	46	ralbidv 3178	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
48		ineq1 4213	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ∩ 𝑤) = (∅ ∩ 𝑤))
49	48	eleq1d 2826	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ 𝑤) ∈ {∅}))
50	49	ralbidv 3178	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ↔ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}))
51		sseq2 4010	. . . . . . . 8 ⊢ (𝑣 = ∅ → (( I ↾ ∅) ⊆ 𝑣 ↔ ( I ↾ ∅) ⊆ ∅))
52		cnveq 5884	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ◡𝑣 = ◡∅)
53	52	eleq1d 2826	. . . . . . . 8 ⊢ (𝑣 = ∅ → (◡𝑣 ∈ {∅} ↔ ◡∅ ∈ {∅}))
54		sseq2 4010	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ ∅))
55	54	rexbidv 3179	. . . . . . . 8 ⊢ (𝑣 = ∅ → (∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅))
56	51, 53, 55	3anbi123d 1438	. . . . . . 7 ⊢ (𝑣 = ∅ → ((( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣) ↔ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅)))
57	47, 50, 56	3anbi123d 1438	. . . . . 6 ⊢ (𝑣 = ∅ → ((∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅))))
58	1, 57	ralsn 4681	. . . . 5 ⊢ (∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅)))
59	27, 33, 44, 58	mpbir3an 1342	. . . 4 ⊢ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣))
60		isust 24212	. . . . 5 ⊢ (∅ ∈ V → ({∅} ∈ (UnifOn‘∅) ↔ ({∅} ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ {∅} ∧ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
61	1, 60	ax-mp 5	. . . 4 ⊢ ({∅} ∈ (UnifOn‘∅) ↔ ({∅} ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ {∅} ∧ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣))))
62	17, 19, 59, 61	mpbir3an 1342	. . 3 ⊢ {∅} ∈ (UnifOn‘∅)
63		snssi 4808	. . 3 ⊢ ({∅} ∈ (UnifOn‘∅) → {{∅}} ⊆ (UnifOn‘∅))
64	62, 63	ax-mp 5	. 2 ⊢ {{∅}} ⊆ (UnifOn‘∅)
65	16, 64	eqssi 4000	1 ⊢ (UnifOn‘∅) = {{∅}}

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626   I cid 5577   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689  ‘cfv 6561  UnifOncust 24208

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ust 24209

This theorem is referenced by:  isusp  24270