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Theorem flimopn 22578
Description: The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
flimopn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐽   𝑥,𝑋

Proof of Theorem flimopn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elflim 22574 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
2 dfss3 3930 . . . 4 (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹)
3 topontop 21516 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 opnneip 21722 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥𝐽𝐴𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
653expb 1117 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
74, 6sylan 583 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
8 eleq1 2901 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝐹𝑥𝐹))
98rspcv 3593 . . . . . . . . 9 (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹𝑥𝐹))
107, 9syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑥𝐽𝐴𝑥)) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹𝑥𝐹))
1110expr 460 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝐴𝑥 → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹𝑥𝐹)))
1211com23 86 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 → (𝐴𝑥𝑥𝐹)))
1312ralrimdva 3179 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 → ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
14 simpr 488 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
153ad3antrrr 729 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
16 simplr 768 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
17 toponuni 21517 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1817ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
1916, 18eleqtrd 2916 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
2019snssd 4715 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
21 eqid 2822 . . . . . . . . . . . . 13 𝐽 = 𝐽
2221neii1 21709 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
234, 22sylan 583 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
2421neiint 21707 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑦 𝐽) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2515, 20, 23, 24syl3anc 1368 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2614, 25mpbid 235 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑦))
27 snssg 4691 . . . . . . . . . 10 (𝐴𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2827ad2antlr 726 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2926, 28mpbird 260 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑦))
3021ntropn 21652 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
3115, 23, 30syl2anc 587 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
32 eleq2 2902 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝐴𝑥𝐴 ∈ ((int‘𝐽)‘𝑦)))
33 eleq1 2901 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝑥𝐹 ↔ ((int‘𝐽)‘𝑦) ∈ 𝐹))
3432, 33imbi12d 348 . . . . . . . . . 10 (𝑥 = ((int‘𝐽)‘𝑦) → ((𝐴𝑥𝑥𝐹) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3534rspcv 3593 . . . . . . . . 9 (((int‘𝐽)‘𝑦) ∈ 𝐽 → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3631, 35syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3729, 36mpid 44 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → ((int‘𝐽)‘𝑦) ∈ 𝐹))
38 simpllr 775 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘𝑋))
3921ntrss2 21660 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4015, 23, 39syl2anc 587 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4123, 18sseqtrrd 3983 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝑋)
42 filss 22456 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝑋 ∧ ((int‘𝐽)‘𝑦) ⊆ 𝑦)) → 𝑦𝐹)
43423exp2 1351 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → (((int‘𝐽)‘𝑦) ∈ 𝐹 → (𝑦𝑋 → (((int‘𝐽)‘𝑦) ⊆ 𝑦𝑦𝐹))))
4443com24 95 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (((int‘𝐽)‘𝑦) ⊆ 𝑦 → (𝑦𝑋 → (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝐹))))
4538, 40, 41, 44syl3c 66 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝐹))
4637, 45syld 47 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → 𝑦𝐹))
4746ralrimdva 3179 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹))
4813, 47impbid 215 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
492, 48syl5bb 286 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
5049pm5.32da 582 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
511, 50bitrd 282 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130  wss 3908  {csn 4539   cuni 4813  cfv 6334  (class class class)co 7140  Topctop 21496  TopOnctopon 21513  intcnt 21620  neicnei 21700  Filcfil 22448   fLim cflim 22537
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-fbas 20086  df-top 21497  df-topon 21514  df-ntr 21623  df-nei 21701  df-fil 22449  df-flim 22542
This theorem is referenced by:  fbflim  22579  flimrest  22586  flimsncls  22589  isflf  22596  cnpflfi  22602  flimfnfcls  22631  alexsublem  22647  cfilfcls  23876  iscmet3lem2  23894
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