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Theorem ustuqtop 24270
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 6906 . . . . . . 7 (𝑝 = 𝑟 → (𝑁𝑝) = (𝑁𝑟))
21eleq2d 2824 . . . . . 6 (𝑝 = 𝑟 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑐 ∈ (𝑁𝑟)))
32cbvralvw 3234 . . . . 5 (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟))
4 eleq1w 2821 . . . . . 6 (𝑐 = 𝑎 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑎 ∈ (𝑁𝑝)))
54raleqbi1dv 3335 . . . . 5 (𝑐 = 𝑎 → (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
63, 5bitr3id 285 . . . 4 (𝑐 = 𝑎 → (∀𝑟𝑐 𝑐 ∈ (𝑁𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
76cbvrabv 3443 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
98ustuqtop0 24264 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
108ustuqtop1 24265 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
118ustuqtop2 24266 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
128ustuqtop3 24267 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
138ustuqtop4 24268 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑥𝑏 𝑎 ∈ (𝑁𝑥))
148ustuqtop5 24269 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
157, 9, 10, 11, 12, 13, 14neiptopreu 23156 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
169feqmptd 6976 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ (𝑁𝑝)))
1716eqeq1d 2736 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18 fvex 6919 . . . . . 6 (𝑁𝑝) ∈ V
1918rgenw 3062 . . . . 5 𝑝𝑋 (𝑁𝑝) ∈ V
20 mpteqb 7034 . . . . 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, 21bitrdi 287 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2322reubidv 3395 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2415, 23mpbid 232 1 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wral 3058  ∃!wreu 3375  {crab 3432  Vcvv 3477  𝒫 cpw 4604  {csn 4630  cmpt 5230  ran crn 5689  cima 5691  cfv 6562  TopOnctopon 22931  neicnei 23120  UnifOncust 24223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-2o 8505  df-en 8984  df-fin 8987  df-fi 9448  df-top 22915  df-topon 22932  df-ntr 23043  df-nei 23121  df-ust 24224
This theorem is referenced by: (None)
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