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Theorem fbflim2 22690
 Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 22689. (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 22689 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦))))
3 topontop 21626 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 725 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 simpr 488 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴𝑋)
6 toponuni 21627 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76ad2antrr 725 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
85, 7eleqtrd 2854 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴 𝐽)
9 eqid 2758 . . . . . . . . 9 𝐽 = 𝐽
109isneip 21818 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
114, 8, 10syl2anc 587 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
12 simpr 488 . . . . . . 7 ((𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))
1311, 12syl6bi 256 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)))
14 r19.29 3181 . . . . . . . 8 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)))
15 pm3.45 624 . . . . . . . . . . 11 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ((𝐴𝑦𝑦𝑛) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛)))
1615imp 410 . . . . . . . . . 10 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛))
17 sstr2 3901 . . . . . . . . . . . . 13 (𝑥𝑦 → (𝑦𝑛𝑥𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦𝑛 → (𝑥𝑦𝑥𝑛))
1918reximdv 3197 . . . . . . . . . . 11 (𝑦𝑛 → (∃𝑥𝐵 𝑥𝑦 → ∃𝑥𝐵 𝑥𝑛))
2019impcom 411 . . . . . . . . . 10 ((∃𝑥𝐵 𝑥𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2221rexlimivw 3206 . . . . . . . 8 (∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2314, 22syl 17 . . . . . . 7 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2423ex 416 . . . . . 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 3117 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
274adantr 484 . . . . . . . . 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 21832 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
3127, 28, 29, 30syl3anc 1368 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32 sseq2 3920 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑥𝑛𝑥𝑦))
3332rexbidv 3221 . . . . . . . . 9 (𝑛 = 𝑦 → (∃𝑥𝐵 𝑥𝑛 ↔ ∃𝑥𝐵 𝑥𝑦))
3433rspcv 3538 . . . . . . . 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 460 . . . . . 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 3118 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3926, 38impbid 215 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
4039pm5.32da 582 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
412, 40bitrd 282 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860  {csn 4525  ∪ cuni 4801  ‘cfv 6340  (class class class)co 7156  fBascfbas 20167  filGencfg 20168  Topctop 21606  TopOnctopon 21623  neicnei 21810   fLim cflim 22647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-fbas 20176  df-fg 20177  df-top 21607  df-topon 21624  df-ntr 21733  df-nei 21811  df-fil 22559  df-flim 22652 This theorem is referenced by: (None)
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