Theorem ustuqtop 22852

Description: For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) is the filter of the neighborhoods of 𝑝 for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

Assertion

Ref	Expression
ustuqtop	⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣,𝑗   𝑗,𝑁,𝑝   𝑣,𝑗,𝑈   𝑗,𝑋

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop

Dummy variables 𝑎 𝑏 𝑐 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6645	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑁‘𝑝) = (𝑁‘𝑟))
2	1	eleq2d 2875	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑐 ∈ (𝑁‘𝑟)))
3	2	cbvralvw 3396	. . . . 5 ⊢ (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟))
4		eleq1w 2872	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ (𝑁‘𝑝)))
5	4	raleqbi1dv 3356	. . . . 5 ⊢ (𝑐 = 𝑎 → (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
6	3, 5	bitr3id 288	. . . 4 ⊢ (𝑐 = 𝑎 → (∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
7	6	cbvrabv 3439	. . 3 ⊢ {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
8		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
9	8	ustuqtop0 22846	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
10	8	ustuqtop1 22847	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
11	8	ustuqtop2 22848	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
12	8	ustuqtop3 22849	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
13	8	ustuqtop4 22850	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑥 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑥))
14	8	ustuqtop5 22851	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 21738	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
16	9	feqmptd 6708	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)))
17	16	eqeq1d 2800	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18		fvex 6658	. . . . . 6 ⊢ (𝑁‘𝑝) ∈ V
19	18	rgenw 3118	. . . . 5 ⊢ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V
20		mpteqb 6764	. . . . 5 ⊢ (∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V → ((𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
21	19, 20	ax-mp 5	. . . 4 ⊢ ((𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
22	17, 21	syl6bb 290	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
23	22	reubidv 3342	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
24	15, 23	mpbid 235	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃!wreu 3108  {crab 3110  Vcvv 3441  𝒫 cpw 4497  {csn 4525   ↦ cmpt 5110  ran crn 5520   “ cima 5522  ‘cfv 6324  TopOnctopon 21515  neicnei 21702  UnifOncust 22805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-fi 8859  df-top 21499  df-topon 21516  df-ntr 21625  df-nei 21703  df-ust 22806

This theorem is referenced by: (None)