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Theorem ustuqtop 22772
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 6666 . . . . . . 7 (𝑝 = 𝑟 → (𝑁𝑝) = (𝑁𝑟))
21eleq2d 2902 . . . . . 6 (𝑝 = 𝑟 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑐 ∈ (𝑁𝑟)))
32cbvralvw 3454 . . . . 5 (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟))
4 eleq1w 2899 . . . . . 6 (𝑐 = 𝑎 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑎 ∈ (𝑁𝑝)))
54raleqbi1dv 3408 . . . . 5 (𝑐 = 𝑎 → (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
63, 5syl5bbr 286 . . . 4 (𝑐 = 𝑎 → (∀𝑟𝑐 𝑐 ∈ (𝑁𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
76cbvrabv 3496 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
98ustuqtop0 22766 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
108ustuqtop1 22767 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
118ustuqtop2 22768 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
128ustuqtop3 22769 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
138ustuqtop4 22770 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑥𝑏 𝑎 ∈ (𝑁𝑥))
148ustuqtop5 22771 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
157, 9, 10, 11, 12, 13, 14neiptopreu 21659 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
169feqmptd 6729 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ (𝑁𝑝)))
1716eqeq1d 2827 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18 fvex 6679 . . . . . 6 (𝑁𝑝) ∈ V
1918rgenw 3154 . . . . 5 𝑝𝑋 (𝑁𝑝) ∈ V
20 mpteqb 6782 . . . . 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 288 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2322reubidv 3394 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝})))
2415, 23mpbid 233 1 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  wral 3142  ∃!wreu 3144  {crab 3146  Vcvv 3499  𝒫 cpw 4541  {csn 4563  cmpt 5142  ran crn 5554  cima 5556  cfv 6351  TopOnctopon 21436  neicnei 21623  UnifOncust 22725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-fin 8505  df-fi 8867  df-top 21420  df-topon 21437  df-ntr 21546  df-nei 21624  df-ust 22726
This theorem is referenced by: (None)
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