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Theorem ustuqtop 22044
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 6189 . . . . . . 7 (𝑝 = 𝑟 → (𝑁𝑝) = (𝑁𝑟))
21eleq2d 2686 . . . . . 6 (𝑝 = 𝑟 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑐 ∈ (𝑁𝑟)))
32cbvralv 3169 . . . . 5 (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟))
4 eleq1 2688 . . . . . 6 (𝑐 = 𝑎 → (𝑐 ∈ (𝑁𝑝) ↔ 𝑎 ∈ (𝑁𝑝)))
54raleqbi1dv 3144 . . . . 5 (𝑐 = 𝑎 → (∀𝑝𝑐 𝑐 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
63, 5syl5bbr 274 . . . 4 (𝑐 = 𝑎 → (∀𝑟𝑐 𝑐 ∈ (𝑁𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)))
76cbvrabv 3197 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑐 𝑐 ∈ (𝑁𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
98ustuqtop0 22038 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
108ustuqtop1 22039 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
118ustuqtop2 22040 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
128ustuqtop3 22041 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
138ustuqtop4 22042 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑥𝑏 𝑎 ∈ (𝑁𝑥))
148ustuqtop5 22043 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
157, 9, 10, 11, 12, 13, 14neiptopreu 20931 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
169feqmptd 6247 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ (𝑁𝑝)))
1716eqeq1d 2623 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝𝑋 ↦ (𝑁𝑝)) = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18 fvex 6199 . . . . . 6 (𝑁𝑝) ∈ V
1918rgenw 2923 . . . . 5 𝑝𝑋 (𝑁𝑝) ∈ V
20 mpteqb 6297 . . . . 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 3124 . 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 1482  wcel 1989  wral 2911  ∃!wreu 2913  {crab 2915  Vcvv 3198  𝒫 cpw 4156  {csn 4175  cmpt 4727  ran crn 5113  cima 5115  cfv 5886  TopOnctopon 20709  neicnei 20895  UnifOncust 21997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-en 7953  df-fin 7956  df-fi 8314  df-top 20693  df-topon 20710  df-ntr 20818  df-nei 20896  df-ust 21998
This theorem is referenced by: (None)
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