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Theorem fbflim2 22000
Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 21999. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbflim.3 𝐹 = (𝑋filGen𝐵)
Assertion
Ref Expression
fbflim2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
Distinct variable groups:   𝑥,𝑛,𝐴   𝐵,𝑛,𝑥   𝑛,𝐽,𝑥   𝑛,𝑋,𝑥   𝑥,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 = (𝑋filGen𝐵)
21fbflim 21999 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦))))
3 topontop 20937 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 697 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 simpr 471 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴𝑋)
6 toponuni 20938 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76ad2antrr 697 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
85, 7eleqtrd 2851 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴 𝐽)
9 eqid 2770 . . . . . . . . 9 𝐽 = 𝐽
109isneip 21129 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
114, 8, 10syl2anc 565 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
12 simpr 471 . . . . . . 7 ((𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))
1311, 12syl6bi 243 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)))
14 r19.29 3219 . . . . . . . 8 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)))
15 pm3.45 600 . . . . . . . . . . 11 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ((𝐴𝑦𝑦𝑛) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛)))
1615imp 393 . . . . . . . . . 10 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛))
17 sstr2 3757 . . . . . . . . . . . . 13 (𝑥𝑦 → (𝑦𝑛𝑥𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦𝑛 → (𝑥𝑦𝑥𝑛))
1918reximdv 3163 . . . . . . . . . . 11 (𝑦𝑛 → (∃𝑥𝐵 𝑥𝑦 → ∃𝑥𝐵 𝑥𝑛))
2019impcom 394 . . . . . . . . . 10 ((∃𝑥𝐵 𝑥𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2221rexlimivw 3176 . . . . . . . 8 (∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2314, 22syl 17 . . . . . . 7 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2423ex 397 . . . . . 6 (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → (∃𝑦𝐽 (𝐴𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛))
2513, 24syl9 77 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑥𝐵 𝑥𝑛)))
2625ralrimdv 3116 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
274adantr 466 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐽 ∈ Top)
28 simprl 746 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦𝐽)
29 simprr 748 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐴𝑦)
30 opnneip 21143 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
3127, 28, 29, 30syl3anc 1475 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32 sseq2 3774 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑥𝑛𝑥𝑦))
3332rexbidv 3199 . . . . . . . . 9 (𝑛 = 𝑦 → (∃𝑥𝐵 𝑥𝑛 ↔ ∃𝑥𝐵 𝑥𝑦))
3433rspcv 3454 . . . . . . . 8 (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦))
3531, 34syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦))
3635expr 444 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦𝐽) → (𝐴𝑦 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦)))
3736com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3837ralrimdva 3117 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3926, 38impbid 202 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
4039pm5.32da 560 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
412, 40bitrd 268 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  wss 3721  {csn 4314   cuni 4572  cfv 6031  (class class class)co 6792  fBascfbas 19948  filGencfg 19949  Topctop 20917  TopOnctopon 20934  neicnei 21121   fLim cflim 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-ntr 21044  df-nei 21122  df-fil 21869  df-flim 21962
This theorem is referenced by: (None)
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