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Theorem fbflim 22512
Description: A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbflim.3 𝐹 = (𝑋filGen𝐵)
Assertion
Ref Expression
fbflim ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝐹,𝑦

Proof of Theorem fbflim
StepHypRef Expression
1 fbflim.3 . . . 4 𝐹 = (𝑋filGen𝐵)
2 fgcl 22414 . . . 4 (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋))
31, 2eqeltrid 2914 . . 3 (𝐵 ∈ (fBas‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
4 flimopn 22511 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
53, 4sylan2 592 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
6 toponss 21463 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
76ad4ant14 748 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
81eleq2i 2901 . . . . . . 7 (𝑥𝐹𝑥 ∈ (𝑋filGen𝐵))
9 elfg 22407 . . . . . . . 8 (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥𝑋 ∧ ∃𝑦𝐵 𝑦𝑥)))
109ad3antlr 727 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥 ∈ (𝑋filGen𝐵) ↔ (𝑥𝑋 ∧ ∃𝑦𝐵 𝑦𝑥)))
118, 10syl5bb 284 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐹 ↔ (𝑥𝑋 ∧ ∃𝑦𝐵 𝑦𝑥)))
127, 11mpbirand 703 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐹 ↔ ∃𝑦𝐵 𝑦𝑥))
1312imbi2d 342 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → ((𝐴𝑥𝑥𝐹) ↔ (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥)))
1413ralbidva 3193 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) ↔ ∀𝑥𝐽 (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥)))
1514pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥))))
165, 15bitrd 280 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  cfv 6348  (class class class)co 7145  fBascfbas 20461  filGencfg 20462  TopOnctopon 21446  Filcfil 22381   fLim cflim 22470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-ntr 21556  df-nei 21634  df-fil 22382  df-flim 22475
This theorem is referenced by:  fbflim2  22513
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