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Theorem flimopn 21998
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 21994 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
2 dfss3 3741 . . . 4 (((nei‘𝐽)‘{𝐴}) ⊆ 𝐹 ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹)
3 topontop 20937 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 705 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 opnneip 21143 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑥𝐽𝐴𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
653expb 1113 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
74, 6sylan 569 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑥𝐽𝐴𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))
8 eleq1 2838 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝐹𝑥𝐹))
98rspcv 3456 . . . . . . . . 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 444 . . . . . . 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 3118 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦 ∈ ((nei‘𝐽)‘{𝐴})𝑦𝐹 → ∀𝑥𝐽 (𝐴𝑥𝑥𝐹)))
14 simpr 471 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
153ad3antrrr 709 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
16 simplr 752 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴𝑋)
17 toponuni 20938 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1817ad3antrrr 709 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
1916, 18eleqtrd 2852 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 𝐽)
2019snssd 4476 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
21 eqid 2771 . . . . . . . . . . . . 13 𝐽 = 𝐽
2221neii1 21130 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
234, 22sylan 569 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 𝐽)
2421neiint 21128 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑦 𝐽) → (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2515, 20, 23, 24syl3anc 1476 . . . . . . . . . 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 4451 . . . . . . . . . 10 (𝐴𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑦) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑦)))
2827ad2antlr 706 . . . . . . . . 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 21073 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
3115, 23, 30syl2anc 573 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ∈ 𝐽)
32 eleq2 2839 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝐴𝑥𝐴 ∈ ((int‘𝐽)‘𝑦)))
33 eleq1 2838 . . . . . . . . . . 11 (𝑥 = ((int‘𝐽)‘𝑦) → (𝑥𝐹 ↔ ((int‘𝐽)‘𝑦) ∈ 𝐹))
3432, 33imbi12d 333 . . . . . . . . . 10 (𝑥 = ((int‘𝐽)‘𝑦) → ((𝐴𝑥𝑥𝐹) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑦) → ((int‘𝐽)‘𝑦) ∈ 𝐹)))
3534rspcv 3456 . . . . . . . . 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 760 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘𝑋))
3921ntrss2 21081 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 𝐽) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4015, 23, 39syl2anc 573 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑦) ⊆ 𝑦)
4123, 18sseqtr4d 3791 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝑋)
42 filss 21876 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (((int‘𝐽)‘𝑦) ∈ 𝐹𝑦𝑋 ∧ ((int‘𝐽)‘𝑦) ⊆ 𝑦)) → 𝑦𝐹)
43423exp2 1447 . . . . . . . . 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 3118 . . . . 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 568 . 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 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  {csn 4317   cuni 4575  cfv 6030  (class class class)co 6795  Topctop 20917  TopOnctopon 20934  intcnt 21041  neicnei 21121  Filcfil 21868   fLim cflim 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-fbas 19957  df-top 20918  df-topon 20935  df-ntr 21044  df-nei 21122  df-fil 21869  df-flim 21962
This theorem is referenced by:  fbflim  21999  flimrest  22006  flimsncls  22009  isflf  22016  cnpflfi  22022  flimfnfcls  22051  alexsublem  22067  cfilfcls  23290  iscmet3lem2  23308
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