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Theorem fbflim2 24001
Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 24000. (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 24000 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦))))
3 topontop 22935 . . . . . . . . 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 22936 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76ad2antrr 726 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
85, 7eleqtrd 2841 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴 𝐽)
9 eqid 2735 . . . . . . . . 9 𝐽 = 𝐽
109isneip 23129 . . . . . . . 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 3112 . . . . . . . 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 4002 . . . . . . . . . . . . 13 (𝑥𝑦 → (𝑦𝑛𝑥𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦𝑛 → (𝑥𝑦𝑥𝑛))
1918reximdv 3168 . . . . . . . . . . 11 (𝑦𝑛 → (∃𝑥𝐵 𝑥𝑦 → ∃𝑥𝐵 𝑥𝑛))
2019impcom 407 . . . . . . . . . 10 ((∃𝑥𝐵 𝑥𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2221rexlimivw 3149 . . . . . . . 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 3150 . . . 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 771 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦𝐽)
29 simprr 773 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐴𝑦)
30 opnneip 23143 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
3127, 28, 29, 30syl3anc 1370 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32 sseq2 4022 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑥𝑛𝑥𝑦))
3332rexbidv 3177 . . . . . . . . 9 (𝑛 = 𝑦 → (∃𝑥𝐵 𝑥𝑛 ↔ ∃𝑥𝐵 𝑥𝑦))
3433rspcv 3618 . . . . . . . 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 3152 . . . 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 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  {csn 4631   cuni 4912  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Topctop 22915  TopOnctopon 22932  neicnei 23121   fLim cflim 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-fil 23870  df-flim 23963
This theorem is referenced by: (None)
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