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Theorem ustuqtop 22251
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 6352 . . . . . . 7 (𝑝 = 𝑟 → (𝑁𝑝) = (𝑁𝑟))
21eleq2d 2825 . . . . . 6 (𝑝 = 𝑟 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑐 ∈ (𝑁𝑟)))
32cbvralv 3310 . . . . 5 (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟))
4 eleq1w 2822 . . . . . 6 (𝑐 = 𝑎 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑎 ∈ (𝑁𝑝)))
54raleqbi1dv 3285 . . . . 5 (𝑐 = 𝑎 → (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
63, 5syl5bbr 274 . . . 4 (𝑐 = 𝑎 → (∀𝑟𝑐 𝑐 ∈ (𝑁𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
76cbvrabv 3339 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
98ustuqtop0 22245 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
108ustuqtop1 22246 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
118ustuqtop2 22247 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
128ustuqtop3 22248 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
138ustuqtop4 22249 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑥𝑏 𝑎 ∈ (𝑁𝑥))
148ustuqtop5 22250 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
157, 9, 10, 11, 12, 13, 14neiptopreu 21139 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
169feqmptd 6411 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ (𝑁𝑝)))
1716eqeq1d 2762 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18 fvex 6362 . . . . . 6 (𝑁𝑝) ∈ V
1918rgenw 3062 . . . . 5 𝑝𝑋 (𝑁𝑝) ∈ V
20 mpteqb 6461 . . . . 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 276 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2322reubidv 3265 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2415, 23mpbid 222 1 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wral 3050  ∃!wreu 3052  {crab 3054  Vcvv 3340  𝒫 cpw 4302  {csn 4321  cmpt 4881  ran crn 5267  cima 5269  cfv 6049  TopOnctopon 20917  neicnei 21103  UnifOncust 22204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-top 20901  df-topon 20918  df-ntr 21026  df-nei 21104  df-ust 22205
This theorem is referenced by: (None)
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