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Theorem flimopn 21702
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 21698 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
2 dfss3 3577 . . . 4 (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹)
3 topontop 20650 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 opnneip 20846 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥𝐽𝐴𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
653expb 1263 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
74, 6sylan 488 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
8 eleq1 2686 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝐹𝑥𝐹))
98rspcv 3294 . . . . . . . . 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 642 . . . . . . 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 2964 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 → ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
14 simpr 477 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
153ad3antrrr 765 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
16 simplr 791 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
17 toponuni 20651 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1817ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
1916, 18eleqtrd 2700 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
2019snssd 4314 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
21 eqid 2621 . . . . . . . . . . . . 13 𝐽 = 𝐽
2221neii1 20833 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
234, 22sylan 488 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
2421neiint 20831 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑦 𝐽) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2515, 20, 23, 24syl3anc 1323 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2614, 25mpbid 222 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑦))
27 snssg 4301 . . . . . . . . . 10 (𝐴𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2827ad2antlr 762 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2926, 28mpbird 247 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑦))
3021ntropn 20776 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
3115, 23, 30syl2anc 692 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
32 eleq2 2687 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝐴𝑥𝐴 ∈ ((int‘𝐽)‘𝑦)))
33 eleq1 2686 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝑥𝐹 ↔ ((int‘𝐽)‘𝑦) ∈ 𝐹))
3432, 33imbi12d 334 . . . . . . . . . 10 (𝑥 = ((int‘𝐽)‘𝑦) → ((𝐴𝑥𝑥𝐹) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3534rspcv 3294 . . . . . . . . 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 798 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘𝑋))
3921ntrss2 20784 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4015, 23, 39syl2anc 692 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4123, 18sseqtr4d 3626 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝑋)
42 filss 21580 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝑋 ∧ ((int‘𝐽)‘𝑦) ⊆ 𝑦)) → 𝑦𝐹)
43423exp2 1282 . . . . . . . . 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 2964 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹))
4813, 47impbid 202 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
492, 48syl5bb 272 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
5049pm5.32da 672 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
511, 50bitrd 268 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559  {csn 4153   cuni 4407  cfv 5852  (class class class)co 6610  Topctop 20630  TopOnctopon 20647  intcnt 20744  neicnei 20824  Filcfil 21572   fLim cflim 21661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-fbas 19675  df-top 20631  df-topon 20648  df-ntr 20747  df-nei 20825  df-fil 21573  df-flim 21666
This theorem is referenced by:  fbflim  21703  flimrest  21710  flimsncls  21713  isflf  21720  cnpflfi  21726  flimfnfcls  21755  alexsublem  21771  cfilfcls  22995  iscmet3lem2  23013
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