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Theorem neifg 34953
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23245. (Contributed by Jeff Hankins, 3-Sep-2009.)
Hypothesis
Ref Expression
neifg.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neifg ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = ((neiβ€˜π½)β€˜π‘†))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝑆   π‘₯,𝑋
Proof of Theorem neifg
Dummy variables 𝑒 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neifg.1 . . . 4 𝑋 = βˆͺ 𝐽
21opnfbas 23245 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∈ (fBasβ€˜π‘‹))
3 fgval 23273 . . 3 ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = {𝑑 ∈ 𝒫 𝑋 ∣ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ…})
42, 3syl 17 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = {𝑑 ∈ 𝒫 𝑋 ∣ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ…})
5 pweq 4594 . . . . . . 7 (𝑑 = 𝑒 β†’ 𝒫 𝑑 = 𝒫 𝑒)
65ineq2d 4192 . . . . . 6 (𝑑 = 𝑒 β†’ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) = ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒))
76neeq1d 2999 . . . . 5 (𝑑 = 𝑒 β†’ (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ… ↔ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ…))
87elrab 3663 . . . 4 (𝑒 ∈ {𝑑 ∈ 𝒫 𝑋 ∣ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ…} ↔ (𝑒 ∈ 𝒫 𝑋 ∧ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ…))
9 velpw 4585 . . . . . . 7 (𝑒 ∈ 𝒫 𝑋 ↔ 𝑒 βŠ† 𝑋)
109a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑒 ∈ 𝒫 𝑋 ↔ 𝑒 βŠ† 𝑋))
11 n0 4326 . . . . . . . 8 (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ… ↔ βˆƒπ‘§ 𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒))
12 elin 3944 . . . . . . . . . 10 (𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ (𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∧ 𝑧 ∈ 𝒫 𝑒))
13 sseq2 3988 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ (𝑆 βŠ† π‘₯ ↔ 𝑆 βŠ† 𝑧))
1413elrab 3663 . . . . . . . . . . 11 (𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ↔ (𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧))
15 velpw 4585 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝑒 ↔ 𝑧 βŠ† 𝑒)
1614, 15anbi12i 627 . . . . . . . . . 10 ((𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∧ 𝑧 ∈ 𝒫 𝑒) ↔ ((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1712, 16bitri 274 . . . . . . . . 9 (𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ ((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1817exbii 1850 . . . . . . . 8 (βˆƒπ‘§ 𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1911, 18bitri 274 . . . . . . 7 (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ… ↔ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
2019a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ… ↔ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒)))
2110, 20anbi12d 631 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((𝑒 ∈ 𝒫 𝑋 ∧ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ…) ↔ (𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))))
22 anass 469 . . . . . . . . 9 (((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ (𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
2322exbii 1850 . . . . . . . 8 (βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§(𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
24 df-rex 3070 . . . . . . . 8 (βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§(𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
2523, 24bitr4i 277 . . . . . . 7 (βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒))
2625anbi2i 623 . . . . . 6 ((𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒)) ↔ (𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
271isnei 22506 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑒 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒))))
28273adant3 1132 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑒 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒))))
2926, 28bitr4id 289 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((𝑒 βŠ† 𝑋 ∧ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒)) ↔ 𝑒 ∈ ((neiβ€˜π½)β€˜π‘†)))
3021, 29bitrd 278 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ ((𝑒 ∈ 𝒫 𝑋 ∧ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ…) ↔ 𝑒 ∈ ((neiβ€˜π½)β€˜π‘†)))
318, 30bitrid 282 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑒 ∈ {𝑑 ∈ 𝒫 𝑋 ∣ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ…} ↔ 𝑒 ∈ ((neiβ€˜π½)β€˜π‘†)))
3231eqrdv 2729 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ {𝑑 ∈ 𝒫 𝑋 ∣ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ…} = ((neiβ€˜π½)β€˜π‘†))
334, 32eqtrd 2771 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = ((neiβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2939  βˆƒwrex 3069  {crab 3418   ∩ cin 3927   βŠ† wss 3928  βˆ…c0 4302  π’« cpw 4580  βˆͺ cuni 4885  β€˜cfv 6516  (class class class)co 7377  fBascfbas 20836  filGencfg 20837  Topctop 22294  neicnei 22500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-fbas 20845  df-fg 20846  df-top 22295  df-nei 22501
This theorem is referenced by: (None)
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