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Theorem fbflim2 23481
Description: A condition for a filter base 𝐡 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23480. (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 23480 . 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 22415 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
43ad2antrr 725 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
5 simpr 486 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
6 toponuni 22416 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76ad2antrr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
85, 7eleqtrd 2836 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐽)
9 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
109isneip 22609 . . . . . . . 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 585 . . . . . . 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 486 . . . . . . 7 ((𝑛 βŠ† βˆͺ 𝐽 ∧ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛)) β†’ βˆƒπ‘¦ ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 βŠ† 𝑛))
1311, 12syl6bi 253 . . . . . 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 3115 . . . . . . . 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 pm3.45 623 . . . . . . . . . . 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 408 . . . . . . . . . 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 3990 . . . . . . . . . . . . 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 3171 . . . . . . . . . . 11 (𝑦 βŠ† 𝑛 β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦 β†’ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛))
2019impcom 409 . . . . . . . . . 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 3152 . . . . . . . 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 414 . . . . . 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 3153 . . . 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 482 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝐽 ∈ Top)
28 simprl 770 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝑦 ∈ 𝐽)
29 simprr 772 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝐴 ∈ 𝑦)
30 opnneip 22623 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
3127, 28, 29, 30syl3anc 1372 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐡 ∈ (fBasβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
32 sseq2 4009 . . . . . . . . . 10 (𝑛 = 𝑦 β†’ (π‘₯ βŠ† 𝑛 ↔ π‘₯ βŠ† 𝑦))
3332rexbidv 3179 . . . . . . . . 9 (𝑛 = 𝑦 β†’ (βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑛 ↔ βˆƒπ‘₯ ∈ 𝐡 π‘₯ βŠ† 𝑦))
3433rspcv 3609 . . . . . . . 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 458 . . . . . 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 3155 . . . 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 580 . 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 279 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  fBascfbas 20932  filGencfg 20933  Topctop 22395  TopOnctopon 22412  neicnei 22601   fLim cflim 23438
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  ax-un 7725
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-topon 22413  df-ntr 22524  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by: (None)
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