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Theorem ustuqtop 22378
Description: For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣𝑈 ↦ (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 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((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.
StepHypRef Expression
1 fveq2 6411 . . . . . . 7 (𝑝 = 𝑟 → (𝑁𝑝) = (𝑁𝑟))
21eleq2d 2864 . . . . . 6 (𝑝 = 𝑟 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑐 ∈ (𝑁𝑟)))
32cbvralv 3354 . . . . 5 (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟))
4 eleq1w 2861 . . . . . 6 (𝑐 = 𝑎 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑎 ∈ (𝑁𝑝)))
54raleqbi1dv 3329 . . . . 5 (𝑐 = 𝑎 → (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
63, 5syl5bbr 277 . . . 4 (𝑐 = 𝑎 → (∀𝑟𝑐 𝑐 ∈ (𝑁𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
76cbvrabv 3383 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
98ustuqtop0 22372 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
108ustuqtop1 22373 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
118ustuqtop2 22374 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
128ustuqtop3 22375 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
138ustuqtop4 22376 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑥𝑏 𝑎 ∈ (𝑁𝑥))
148ustuqtop5 22377 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
157, 9, 10, 11, 12, 13, 14neiptopreu 21266 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
169feqmptd 6474 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ (𝑁𝑝)))
1716eqeq1d 2801 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18 fvex 6424 . . . . . 6 (𝑁𝑝) ∈ V
1918rgenw 3105 . . . . 5 𝑝𝑋 (𝑁𝑝) ∈ V
20 mpteqb 6524 . . . . 5 (∀𝑝𝑋 (𝑁𝑝) ∈ V → ((𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2119, 20ax-mp 5 . . . 4 ((𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2217, 21syl6bb 279 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2322reubidv 3309 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2415, 23mpbid 224 1 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3089  ∃!wreu 3091  {crab 3093  Vcvv 3385  𝒫 cpw 4349  {csn 4368  cmpt 4922  ran crn 5313  cima 5315  cfv 6101  TopOnctopon 21043  neicnei 21230  UnifOncust 22331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-fin 8199  df-fi 8559  df-top 21027  df-topon 21044  df-ntr 21153  df-nei 21231  df-ust 22332
This theorem is referenced by: (None)
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