Theorem ust0 23279

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 5226	. . . . . . . 8 ⊢ ∅ ∈ V
2		isust 23263	. . . . . . . 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 1143	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ 𝒫 (∅ × ∅))
5		0xp 5675	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
6	5	pweqi 4548	. . . . . . 7 ⊢ 𝒫 (∅ × ∅) = 𝒫 ∅
7		pw0 4742	. . . . . . 7 ⊢ 𝒫 ∅ = {∅}
8	6, 7	eqtri 2766	. . . . . 6 ⊢ 𝒫 (∅ × ∅) = {∅}
9	4, 8	sseqtrdi 3967	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ {∅})
10		ustbasel 23266	. . . . . . 7 ⊢ (𝑢 ∈ (UnifOn‘∅) → (∅ × ∅) ∈ 𝑢)
11	5, 10	eqeltrrid 2844	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → ∅ ∈ 𝑢)
12	11	snssd 4739	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → {∅} ⊆ 𝑢)
13	9, 12	eqssd 3934	. . . 4 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 = {∅})
14		velsn 4574	. . . 4 ⊢ (𝑢 ∈ {{∅}} ↔ 𝑢 = {∅})
15	13, 14	sylibr 233	. . 3 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ∈ {{∅}})
16	15	ssriv 3921	. 2 ⊢ (UnifOn‘∅) ⊆ {{∅}}
17	8	eqimss2i 3976	. . . 4 ⊢ {∅} ⊆ 𝒫 (∅ × ∅)
18	1	snid 4594	. . . . 5 ⊢ ∅ ∈ {∅}
19	5, 18	eqeltri 2835	. . . 4 ⊢ (∅ × ∅) ∈ {∅}
20	18	a1i 11	. . . . . 6 ⊢ (∅ ⊆ ∅ → ∅ ∈ {∅})
21	8	raleqi 3337	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ {∅} (∅ ⊆ 𝑤 → 𝑤 ∈ {∅}))
22		sseq2 3943	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (∅ ⊆ 𝑤 ↔ ∅ ⊆ ∅))
23		eleq1 2826	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
24	22, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅})))
25	1, 24	ralsn 4614	. . . . . . 7 ⊢ (∀𝑤 ∈ {∅} (∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
26	21, 25	bitri 274	. . . . . 6 ⊢ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
27	20, 26	mpbir 230	. . . . 5 ⊢ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅})
28		inidm 4149	. . . . . . 7 ⊢ (∅ ∩ ∅) = ∅
29	28, 18	eqeltri 2835	. . . . . 6 ⊢ (∅ ∩ ∅) ∈ {∅}
30		ineq2 4137	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∅ ∩ 𝑤) = (∅ ∩ ∅))
31	30	eleq1d 2823	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅}))
32	1, 31	ralsn 4614	. . . . . 6 ⊢ (∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅})
33	29, 32	mpbir 230	. . . . 5 ⊢ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}
34		res0 5884	. . . . . . 7 ⊢ ( I ↾ ∅) = ∅
35	34	eqimssi 3975	. . . . . 6 ⊢ ( I ↾ ∅) ⊆ ∅
36		cnv0 6033	. . . . . . 7 ⊢ ◡∅ = ∅
37	36, 18	eqeltri 2835	. . . . . 6 ⊢ ◡∅ ∈ {∅}
38		0trrel 14620	. . . . . . 7 ⊢ (∅ ∘ ∅) ⊆ ∅
39		id 22	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → 𝑤 = ∅)
40	39, 39	coeq12d 5762	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∘ 𝑤) = (∅ ∘ ∅))
41	40	sseq1d 3948	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((𝑤 ∘ 𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅))
42	1, 41	rexsn 4615	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅)
43	38, 42	mpbir 230	. . . . . 6 ⊢ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅
44	35, 37, 43	3pm3.2i 1337	. . . . 5 ⊢ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅)
45		sseq1 3942	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ⊆ 𝑤 ↔ ∅ ⊆ 𝑤))
46	45	imbi1d 341	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
47	46	ralbidv 3120	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
48		ineq1 4136	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ∩ 𝑤) = (∅ ∩ 𝑤))
49	48	eleq1d 2823	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ 𝑤) ∈ {∅}))
50	49	ralbidv 3120	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ↔ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}))
51		sseq2 3943	. . . . . . . 8 ⊢ (𝑣 = ∅ → (( I ↾ ∅) ⊆ 𝑣 ↔ ( I ↾ ∅) ⊆ ∅))
52		cnveq 5771	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ◡𝑣 = ◡∅)
53	52	eleq1d 2823	. . . . . . . 8 ⊢ (𝑣 = ∅ → (◡𝑣 ∈ {∅} ↔ ◡∅ ∈ {∅}))
54		sseq2 3943	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ ∅))
55	54	rexbidv 3225	. . . . . . . 8 ⊢ (𝑣 = ∅ → (∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅))
56	51, 53, 55	3anbi123d 1434	. . . . . . 7 ⊢ (𝑣 = ∅ → ((( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣) ↔ (( I ↾ ∅) ⊆ ∅ ∧ ◡∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ ∅)))
57	47, 50, 56	3anbi123d 1434	. . . . . 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 4614	. . . . 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 1339	. . . 4 ⊢ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣 ∧ ◡𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤 ∘ 𝑤) ⊆ 𝑣))
60		isust 23263	. . . . 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 1339	. . 3 ⊢ {∅} ∈ (UnifOn‘∅)
63		snssi 4738	. . 3 ⊢ ({∅} ∈ (UnifOn‘∅) → {{∅}} ⊆ (UnifOn‘∅))
64	62, 63	ax-mp 5	. 2 ⊢ {{∅}} ⊆ (UnifOn‘∅)
65	16, 64	eqssi 3933	1 ⊢ (UnifOn‘∅) = {{∅}}

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558   I cid 5479   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   ∘ ccom 5584  ‘cfv 6418  UnifOncust 23259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ust 23260

This theorem is referenced by:  isusp  23321