Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimopn Structured version   Visualization version   GIF version

Theorem flimopn 22187
 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 22183 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
2 dfss3 3809 . . . 4 (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹)
3 topontop 21125 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 716 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 opnneip 21331 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥𝐽𝐴𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
653expb 1110 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
74, 6sylan 575 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
8 eleq1 2846 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝐹𝑥𝐹))
98rspcv 3506 . . . . . . . . 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 450 . . . . . . 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 3150 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 → ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
14 simpr 479 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
153ad3antrrr 720 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
16 simplr 759 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
17 toponuni 21126 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1817ad3antrrr 720 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
1916, 18eleqtrd 2860 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
2019snssd 4571 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
21 eqid 2777 . . . . . . . . . . . . 13 𝐽 = 𝐽
2221neii1 21318 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
234, 22sylan 575 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
2421neiint 21316 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑦 𝐽) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2515, 20, 23, 24syl3anc 1439 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2614, 25mpbid 224 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑦))
27 snssg 4547 . . . . . . . . . 10 (𝐴𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2827ad2antlr 717 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2926, 28mpbird 249 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑦))
3021ntropn 21261 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
3115, 23, 30syl2anc 579 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
32 eleq2 2847 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝐴𝑥𝐴 ∈ ((int‘𝐽)‘𝑦)))
33 eleq1 2846 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝑥𝐹 ↔ ((int‘𝐽)‘𝑦) ∈ 𝐹))
3432, 33imbi12d 336 . . . . . . . . . 10 (𝑥 = ((int‘𝐽)‘𝑦) → ((𝐴𝑥𝑥𝐹) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3534rspcv 3506 . . . . . . . . 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 766 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘𝑋))
3921ntrss2 21269 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4015, 23, 39syl2anc 579 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4123, 18sseqtr4d 3860 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝑋)
42 filss 22065 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝑋 ∧ ((int‘𝐽)‘𝑦) ⊆ 𝑦)) → 𝑦𝐹)
43423exp2 1416 . . . . . . . . 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 3150 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑥𝐽 (𝐴𝑥𝑥𝐹) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹))
4813, 47impbid 204 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
492, 48syl5bb 275 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
5049pm5.32da 574 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))
511, 50bitrd 271 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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∀wral 3089   ⊆ wss 3791  {csn 4397  ∪ cuni 4671  ‘cfv 6135  (class class class)co 6922  Topctop 21105  TopOnctopon 21122  intcnt 21229  neicnei 21309  Filcfil 22057   fLim cflim 22146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-fbas 20139  df-top 21106  df-topon 21123  df-ntr 21232  df-nei 21310  df-fil 22058  df-flim 22151 This theorem is referenced by:  fbflim  22188  flimrest  22195  flimsncls  22198  isflf  22205  cnpflfi  22211  flimfnfcls  22240  alexsublem  22256  cfilfcls  23480  iscmet3lem2  23498
 Copyright terms: Public domain W3C validator