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Theorem neifg 35256
Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23346. (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 23346 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∈ (fBasβ€˜π‘‹))
3 fgval 23374 . . 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 4617 . . . . . . 7 (𝑑 = 𝑒 β†’ 𝒫 𝑑 = 𝒫 𝑒)
65ineq2d 4213 . . . . . 6 (𝑑 = 𝑒 β†’ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) = ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒))
76neeq1d 3001 . . . . 5 (𝑑 = 𝑒 β†’ (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑑) β‰  βˆ… ↔ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ…))
87elrab 3684 . . . 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 4608 . . . . . . 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 4347 . . . . . . . 8 (({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) β‰  βˆ… ↔ βˆƒπ‘§ 𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒))
12 elin 3965 . . . . . . . . . 10 (𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ (𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∧ 𝑧 ∈ 𝒫 𝑒))
13 sseq2 4009 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ (𝑆 βŠ† π‘₯ ↔ 𝑆 βŠ† 𝑧))
1413elrab 3684 . . . . . . . . . . 11 (𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ↔ (𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧))
15 velpw 4608 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝑒 ↔ 𝑧 βŠ† 𝑒)
1614, 15anbi12i 628 . . . . . . . . . 10 ((𝑧 ∈ {π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∧ 𝑧 ∈ 𝒫 𝑒) ↔ ((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1712, 16bitri 275 . . . . . . . . 9 (𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ ((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1817exbii 1851 . . . . . . . 8 (βˆƒπ‘§ 𝑧 ∈ ({π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯} ∩ 𝒫 𝑒) ↔ βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒))
1911, 18bitri 275 . . . . . . 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 632 . . . . 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 470 . . . . . . . . 9 (((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ (𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
2322exbii 1851 . . . . . . . 8 (βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§(𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
24 df-rex 3072 . . . . . . . 8 (βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§(𝑧 ∈ 𝐽 ∧ (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒)))
2523, 24bitr4i 278 . . . . . . 7 (βˆƒπ‘§((𝑧 ∈ 𝐽 ∧ 𝑆 βŠ† 𝑧) ∧ 𝑧 βŠ† 𝑒) ↔ βˆƒπ‘§ ∈ 𝐽 (𝑆 βŠ† 𝑧 ∧ 𝑧 βŠ† 𝑒))
2625anbi2i 624 . . . . . 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 22607 . . . . . . 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 1133 . . . . . 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 290 . . . . 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 279 . . . 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 283 . . 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 2731 . 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 2773 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  {crab 3433   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  fBascfbas 20932  filGencfg 20933  Topctop 22395  neicnei 22601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-fg 20942  df-top 22396  df-nei 22602
This theorem is referenced by: (None)
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