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Theorem fbflim2 23986
Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23985. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbflim.3 𝐹 = (𝑋filGen𝐵)
Assertion
Ref Expression
fbflim2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
Distinct variable groups:   𝑥,𝑛,𝐴   𝐵,𝑛,𝑥   𝑛,𝐽,𝑥   𝑛,𝑋,𝑥   𝑥,𝐹
Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem fbflim2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fbflim.3 . . 3 𝐹 = (𝑋filGen𝐵)
21fbflim 23985 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦))))
3 topontop 22920 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 726 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 simpr 484 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴𝑋)
6 toponuni 22921 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76ad2antrr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
85, 7eleqtrd 2842 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴 𝐽)
9 eqid 2736 . . . . . . . . 9 𝐽 = 𝐽
109isneip 23114 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
114, 8, 10syl2anc 584 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
12 simpr 484 . . . . . . 7 ((𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))
1311, 12biimtrdi 253 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)))
14 r19.29 3113 . . . . . . . 8 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)))
15 pm3.45 622 . . . . . . . . . . 11 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ((𝐴𝑦𝑦𝑛) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛)))
1615imp 406 . . . . . . . . . 10 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛))
17 sstr2 3989 . . . . . . . . . . . . 13 (𝑥𝑦 → (𝑦𝑛𝑥𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦𝑛 → (𝑥𝑦𝑥𝑛))
1918reximdv 3169 . . . . . . . . . . 11 (𝑦𝑛 → (∃𝑥𝐵 𝑥𝑦 → ∃𝑥𝐵 𝑥𝑛))
2019impcom 407 . . . . . . . . . 10 ((∃𝑥𝐵 𝑥𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2221rexlimivw 3150 . . . . . . . 8 (∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2314, 22syl 17 . . . . . . 7 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2423ex 412 . . . . . 6 (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → (∃𝑦𝐽 (𝐴𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛))
2513, 24syl9 77 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑥𝐵 𝑥𝑛)))
2625ralrimdv 3151 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
274adantr 480 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐽 ∈ Top)
28 simprl 770 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦𝐽)
29 simprr 772 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐴𝑦)
30 opnneip 23128 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
3127, 28, 29, 30syl3anc 1372 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32 sseq2 4009 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑥𝑛𝑥𝑦))
3332rexbidv 3178 . . . . . . . . 9 (𝑛 = 𝑦 → (∃𝑥𝐵 𝑥𝑛 ↔ ∃𝑥𝐵 𝑥𝑦))
3433rspcv 3617 . . . . . . . 8 (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦))
3531, 34syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦))
3635expr 456 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦𝐽) → (𝐴𝑦 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∃𝑥𝐵 𝑥𝑦)))
3736com23 86 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ 𝑦𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3837ralrimdva 3153 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3926, 38impbid 212 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
4039pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
412, 40bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  wss 3950  {csn 4625   cuni 4906  cfv 6560  (class class class)co 7432  fBascfbas 21353  filGencfg 21354  Topctop 22900  TopOnctopon 22917  neicnei 23106   fLim cflim 23943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-fg 21363  df-top 22901  df-topon 22918  df-ntr 23029  df-nei 23107  df-fil 23855  df-flim 23948
This theorem is referenced by: (None)
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