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Theorem ustuqtop 23972
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 6891 . . . . . . 7 (𝑝 = π‘Ÿ β†’ (π‘β€˜π‘) = (π‘β€˜π‘Ÿ))
21eleq2d 2818 . . . . . 6 (𝑝 = π‘Ÿ β†’ (𝑐 ∈ (π‘β€˜π‘) ↔ 𝑐 ∈ (π‘β€˜π‘Ÿ)))
32cbvralvw 3233 . . . . 5 (βˆ€π‘ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘) ↔ βˆ€π‘Ÿ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘Ÿ))
4 eleq1w 2815 . . . . . 6 (𝑐 = π‘Ž β†’ (𝑐 ∈ (π‘β€˜π‘) ↔ π‘Ž ∈ (π‘β€˜π‘)))
54raleqbi1dv 3332 . . . . 5 (𝑐 = π‘Ž β†’ (βˆ€π‘ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘) ↔ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
63, 5bitr3id 285 . . . 4 (𝑐 = π‘Ž β†’ (βˆ€π‘Ÿ ∈ 𝑐 𝑐 ∈ (π‘β€˜π‘Ÿ) ↔ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)))
76cbvrabv 3441 . . 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 23966 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
108ustuqtop1 23967 . . 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 23968 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
128ustuqtop3 23969 . . 3 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
138ustuqtop4 23970 . . 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 23971 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
157, 9, 10, 11, 12, 13, 14neiptopreu 22858 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
169feqmptd 6960 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ (π‘β€˜π‘)))
1716eqeq1d 2733 . . . 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 6904 . . . . . 6 (π‘β€˜π‘) ∈ V
1918rgenw 3064 . . . . 5 βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) ∈ V
20 mpteqb 7017 . . . . 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 287 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ βˆ€π‘ ∈ 𝑋 (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝})))
2322reubidv 3393 . 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 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒ!wreu 3373  {crab 3431  Vcvv 3473  π’« cpw 4602  {csn 4628   ↦ cmpt 5231  ran crn 5677   β€œ cima 5679  β€˜cfv 6543  TopOnctopon 22633  neicnei 22822  UnifOncust 23925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7860  df-1o 8470  df-er 8707  df-en 8944  df-fin 8947  df-fi 9410  df-top 22617  df-topon 22634  df-ntr 22745  df-nei 22823  df-ust 23926
This theorem is referenced by: (None)
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