Theorem ust0 24114

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 5265	. . . . . . . 8 ⊢ ∅ ∈ V
2		isust 24098	. . . . . . . 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 1145	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ 𝒫 (∅ × ∅))
5		0xp 5740	. . . . . . . 8 ⊢ (∅ × ∅) = ∅
6	5	pweqi 4582	. . . . . . 7 ⊢ 𝒫 (∅ × ∅) = 𝒫 ∅
7		pw0 4779	. . . . . . 7 ⊢ 𝒫 ∅ = {∅}
8	6, 7	eqtri 2753	. . . . . 6 ⊢ 𝒫 (∅ × ∅) = {∅}
9	4, 8	sseqtrdi 3990	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ {∅})
10		ustbasel 24101	. . . . . . 7 ⊢ (𝑢 ∈ (UnifOn‘∅) → (∅ × ∅) ∈ 𝑢)
11	5, 10	eqeltrrid 2834	. . . . . 6 ⊢ (𝑢 ∈ (UnifOn‘∅) → ∅ ∈ 𝑢)
12	11	snssd 4776	. . . . 5 ⊢ (𝑢 ∈ (UnifOn‘∅) → {∅} ⊆ 𝑢)
13	9, 12	eqssd 3967	. . . 4 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 = {∅})
14		velsn 4608	. . . 4 ⊢ (𝑢 ∈ {{∅}} ↔ 𝑢 = {∅})
15	13, 14	sylibr 234	. . 3 ⊢ (𝑢 ∈ (UnifOn‘∅) → 𝑢 ∈ {{∅}})
16	15	ssriv 3953	. 2 ⊢ (UnifOn‘∅) ⊆ {{∅}}
17	8	eqimss2i 4011	. . . 4 ⊢ {∅} ⊆ 𝒫 (∅ × ∅)
18	1	snid 4629	. . . . 5 ⊢ ∅ ∈ {∅}
19	5, 18	eqeltri 2825	. . . 4 ⊢ (∅ × ∅) ∈ {∅}
20	18	a1i 11	. . . . . 6 ⊢ (∅ ⊆ ∅ → ∅ ∈ {∅})
21	8	raleqi 3299	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ {∅} (∅ ⊆ 𝑤 → 𝑤 ∈ {∅}))
22		sseq2 3976	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (∅ ⊆ 𝑤 ↔ ∅ ⊆ ∅))
23		eleq1 2817	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
24	22, 23	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((∅ ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅})))
25	1, 24	ralsn 4648	. . . . . . 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 4193	. . . . . . 7 ⊢ (∅ ∩ ∅) = ∅
29	28, 18	eqeltri 2825	. . . . . 6 ⊢ (∅ ∩ ∅) ∈ {∅}
30		ineq2 4180	. . . . . . . 8 ⊢ (𝑤 = ∅ → (∅ ∩ 𝑤) = (∅ ∩ ∅))
31	30	eleq1d 2814	. . . . . . 7 ⊢ (𝑤 = ∅ → ((∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅}))
32	1, 31	ralsn 4648	. . . . . 6 ⊢ (∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅})
33	29, 32	mpbir 231	. . . . 5 ⊢ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}
34		res0 5957	. . . . . . 7 ⊢ ( I ↾ ∅) = ∅
35	34	eqimssi 4010	. . . . . 6 ⊢ ( I ↾ ∅) ⊆ ∅
36		cnv0 6116	. . . . . . 7 ⊢ ◡∅ = ∅
37	36, 18	eqeltri 2825	. . . . . 6 ⊢ ◡∅ ∈ {∅}
38		0trrel 14954	. . . . . . 7 ⊢ (∅ ∘ ∅) ⊆ ∅
39		id 22	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → 𝑤 = ∅)
40	39, 39	coeq12d 5831	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 ∘ 𝑤) = (∅ ∘ ∅))
41	40	sseq1d 3981	. . . . . . . 8 ⊢ (𝑤 = ∅ → ((𝑤 ∘ 𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅))
42	1, 41	rexsn 4649	. . . . . . 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 3975	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ⊆ 𝑤 ↔ ∅ ⊆ 𝑤))
46	45	imbi1d 341	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ (∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
47	46	ralbidv 3157	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣 ⊆ 𝑤 → 𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤 → 𝑤 ∈ {∅})))
48		ineq1 4179	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝑣 ∩ 𝑤) = (∅ ∩ 𝑤))
49	48	eleq1d 2814	. . . . . . . 8 ⊢ (𝑣 = ∅ → ((𝑣 ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ 𝑤) ∈ {∅}))
50	49	ralbidv 3157	. . . . . . 7 ⊢ (𝑣 = ∅ → (∀𝑤 ∈ {∅} (𝑣 ∩ 𝑤) ∈ {∅} ↔ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}))
51		sseq2 3976	. . . . . . . 8 ⊢ (𝑣 = ∅ → (( I ↾ ∅) ⊆ 𝑣 ↔ ( I ↾ ∅) ⊆ ∅))
52		cnveq 5840	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ◡𝑣 = ◡∅)
53	52	eleq1d 2814	. . . . . . . 8 ⊢ (𝑣 = ∅ → (◡𝑣 ∈ {∅} ↔ ◡∅ ∈ {∅}))
54		sseq2 3976	. . . . . . . . 9 ⊢ (𝑣 = ∅ → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ ∅))
55	54	rexbidv 3158	. . . . . . . 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 4648	. . . . 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 24098	. . . . 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 4775	. . 3 ⊢ ({∅} ∈ (UnifOn‘∅) → {{∅}} ⊆ (UnifOn‘∅))
64	62, 63	ax-mp 5	. 2 ⊢ {{∅}} ⊆ (UnifOn‘∅)
65	16, 64	eqssi 3966	1 ⊢ (UnifOn‘∅) = {{∅}}

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592   I cid 5535   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  ‘cfv 6514  UnifOncust 24094

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-ust 24095

This theorem is referenced by:  isusp  24156