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Theorem fbflim2 23701
Description: A condition for a filter base 𝐡 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23700. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbflim.3 𝐹 = (𝑋filGen𝐡)
Assertion
Ref Expression
fbflim2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)))
Distinct variable groups:   π‘₯,𝑛,𝐴   𝐡,𝑛,π‘₯   𝑛,𝐽,π‘₯   𝑛,𝑋,π‘₯   π‘₯,𝐹
Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem fbflim2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fbflim.3 . . 3 𝐹 = (𝑋filGen𝐡)
21fbflim 23700 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦))))
3 topontop 22635 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
43ad2antrr 724 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
5 simpr 485 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
6 toponuni 22636 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76ad2antrr 724 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
85, 7eleqtrd 2835 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐽)
9 eqid 2732 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
109isneip 22829 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 ∈ βˆͺ 𝐽) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ (𝑛 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛))))
114, 8, 10syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ (𝑛 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛))))
12 simpr 485 . . . . . . 7 ((𝑛 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛))
1311, 12syl6bi 252 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)))
14 r19.29 3114 . . . . . . . 8 ((βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘¦ ∈ 𝐽 ((𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)))
15 pm3.45 622 . . . . . . . . . . 11 ((𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) β†’ ((𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛) β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦 ∧ 𝑦 βŠ† 𝑛)))
1615imp 407 . . . . . . . . . 10 (((𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦 ∧ 𝑦 βŠ† 𝑛))
17 sstr2 3989 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝑦 β†’ (𝑦 βŠ† 𝑛 β†’ π‘₯ βŠ† 𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦 βŠ† 𝑛 β†’ (π‘₯ βŠ† 𝑦 β†’ π‘₯ βŠ† 𝑛))
1918reximdv 3170 . . . . . . . . . . 11 (𝑦 βŠ† 𝑛 β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛))
2019impcom 408 . . . . . . . . . 10 ((βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦 ∧ 𝑦 βŠ† 𝑛) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)
2221rexlimivw 3151 . . . . . . . 8 (βˆƒπ‘¦ ∈ 𝐽 ((𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)
2314, 22syl 17 . . . . . . 7 ((βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)
2423ex 413 . . . . . 6 (βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) β†’ (βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛))
2513, 24syl9 77 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) β†’ (𝑛 ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)))
2625ralrimdv 3152 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) β†’ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛))
274adantr 481 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝐽 ∈ Top)
28 simprl 769 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝑦 ∈ 𝐽)
29 simprr 771 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝐴 ∈ 𝑦)
30 opnneip 22843 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
3127, 28, 29, 30syl3anc 1371 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
32 sseq2 4008 . . . . . . . . . 10 (𝑛 = 𝑦 β†’ (π‘₯ βŠ† 𝑛 ↔ π‘₯ βŠ† 𝑦))
3332rexbidv 3178 . . . . . . . . 9 (𝑛 = 𝑦 β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 ↔ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦))
3433rspcv 3608 . . . . . . . 8 (𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦))
3531, 34syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦))
3635expr 457 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐽) β†’ (𝐴 ∈ 𝑦 β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦)))
3736com23 86 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐽) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 β†’ (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦)))
3837ralrimdva 3154 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 β†’ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦)))
3926, 38impbid 211 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦) ↔ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛))
4039pm5.32da 579 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) β†’ ((𝐴 ∈ 𝑋 ∧ βˆ€π‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦)) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)))
412, 40bitrd 278 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘› ∈ ((neiβ€˜π½)β€˜{𝐴})βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  fBascfbas 21132  filGencfg 21133  Topctop 22615  TopOnctopon 22632  neicnei 22821   fLim cflim 23658
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822  df-fil 23570  df-flim 23663
This theorem is referenced by: (None)
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