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Theorem ustuqtop 23971
Description: For a given uniform structure π‘ˆ on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {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 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
Assertion
Ref Expression
ustuqtop (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝}))
Distinct variable groups:   𝑣,𝑝,π‘ˆ   𝑋,𝑝,𝑣,𝑗   𝑗,𝑁,𝑝   𝑣,𝑗,π‘ˆ   𝑗,𝑋
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop
Dummy variables π‘Ž 𝑏 𝑐 π‘Ÿ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6890 . . . . . . 7 (𝑝 = π‘Ÿ β†’ (π‘β€˜π‘) = (π‘β€˜π‘Ÿ))
21eleq2d 2817 . . . . . 6 (𝑝 = π‘Ÿ β†’ (𝑐 ∈ (π‘β€˜π‘) ↔ 𝑐 ∈ (π‘β€˜π‘Ÿ)))
32cbvralvw 3232 . . . . 5 (βˆ€π‘ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘) ↔ βˆ€π‘Ÿ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘Ÿ))
4 eleq1w 2814 . . . . . 6 (𝑐 = π‘Ž β†’ (𝑐 ∈ (π‘β€˜π‘) ↔ π‘Ž ∈ (π‘β€˜π‘)))
54raleqbi1dv 3331 . . . . 5 (𝑐 = π‘Ž β†’ (βˆ€π‘ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘) ↔ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
63, 5bitr3id 284 . . . 4 (𝑐 = π‘Ž β†’ (βˆ€π‘Ÿ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘Ÿ) ↔ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
76cbvrabv 3440 . . 3 {𝑐 ∈ 𝒫 𝑋 ∣ βˆ€π‘Ÿ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘Ÿ)} = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
8 utopustuq.1 . . . 4 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
98ustuqtop0 23965 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
108ustuqtop1 23966 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
118ustuqtop2 23967 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
128ustuqtop3 23968 . . 3 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
138ustuqtop4 23969 . . 3 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘ ∈ (π‘β€˜π‘)βˆ€π‘₯ ∈ 𝑏 π‘Ž ∈ (π‘β€˜π‘₯))
148ustuqtop5 23970 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
157, 9, 10, 11, 12, 13, 14neiptopreu 22857 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
169feqmptd 6959 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ (π‘β€˜π‘)))
1716eqeq1d 2732 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (π‘β€˜π‘)) = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))))
18 fvex 6903 . . . . . 6 (π‘β€˜π‘) ∈ V
1918rgenw 3063 . . . . 5 βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) ∈ V
20 mpteqb 7016 . . . . 5 (βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) ∈ V β†’ ((𝑝 ∈ 𝑋 ↦ (π‘β€˜π‘)) = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝})))
2119, 20ax-mp 5 . . . 4 ((𝑝 ∈ 𝑋 ↦ (π‘β€˜π‘)) = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝}))
2217, 21bitrdi 286 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝})))
2322reubidv 3392 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝})))
2415, 23mpbid 231 1 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒ!wreu 3372  {crab 3430  Vcvv 3472  π’« cpw 4601  {csn 4627   ↦ cmpt 5230  ran crn 5676   β€œ cima 5678  β€˜cfv 6542  TopOnctopon 22632  neicnei 22821  UnifOncust 23924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822  df-ust 23925
This theorem is referenced by: (None)
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