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Theorem fbflim2 22060
Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 22059. (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 22059 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦))))
3 topontop 20997 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
43ad2antrr 717 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
5 simpr 477 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴𝑋)
6 toponuni 20998 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76ad2antrr 717 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
85, 7eleqtrd 2846 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → 𝐴 𝐽)
9 eqid 2765 . . . . . . . . 9 𝐽 = 𝐽
109isneip 21189 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
114, 8, 10syl2anc 579 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))))
12 simpr 477 . . . . . . 7 ((𝑛 𝐽 ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛))
1311, 12syl6bi 244 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)))
14 r19.29 3219 . . . . . . . 8 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)))
15 pm3.45 615 . . . . . . . . . . 11 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ((𝐴𝑦𝑦𝑛) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛)))
1615imp 395 . . . . . . . . . 10 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → (∃𝑥𝐵 𝑥𝑦𝑦𝑛))
17 sstr2 3768 . . . . . . . . . . . . 13 (𝑥𝑦 → (𝑦𝑛𝑥𝑛))
1817com12 32 . . . . . . . . . . . 12 (𝑦𝑛 → (𝑥𝑦𝑥𝑛))
1918reximdv 3162 . . . . . . . . . . 11 (𝑦𝑛 → (∃𝑥𝐵 𝑥𝑦 → ∃𝑥𝐵 𝑥𝑛))
2019impcom 396 . . . . . . . . . 10 ((∃𝑥𝐵 𝑥𝑦𝑦𝑛) → ∃𝑥𝐵 𝑥𝑛)
2116, 20syl 17 . . . . . . . . 9 (((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2221rexlimivw 3176 . . . . . . . 8 (∃𝑦𝐽 ((𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2314, 22syl 17 . . . . . . 7 ((∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ∧ ∃𝑦𝐽 (𝐴𝑦𝑦𝑛)) → ∃𝑥𝐵 𝑥𝑛)
2423ex 401 . . . . . 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 3115 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
274adantr 472 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐽 ∈ Top)
28 simprl 787 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦𝐽)
29 simprr 789 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝐴𝑦)
30 opnneip 21203 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
3127, 28, 29, 30syl3anc 1490 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) ∧ (𝑦𝐽𝐴𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32 sseq2 3787 . . . . . . . . . 10 (𝑛 = 𝑦 → (𝑥𝑛𝑥𝑦))
3332rexbidv 3199 . . . . . . . . 9 (𝑛 = 𝑦 → (∃𝑥𝐵 𝑥𝑛 ↔ ∃𝑥𝐵 𝑥𝑦))
3433rspcv 3457 . . . . . . . 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 448 . . . . . 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 3116 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛 → ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)))
3926, 38impbid 203 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴𝑋) → (∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛))
4039pm5.32da 574 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴𝑋 ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑥𝐵 𝑥𝑦)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))
412, 40bitrd 270 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3732  {csn 4334   cuni 4594  cfv 6068  (class class class)co 6842  fBascfbas 20007  filGencfg 20008  Topctop 20977  TopOnctopon 20994  neicnei 21181   fLim cflim 22017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-fbas 20016  df-fg 20017  df-top 20978  df-topon 20995  df-ntr 21104  df-nei 21182  df-fil 21929  df-flim 22022
This theorem is referenced by: (None)
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