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Theorem flimclsi 22875
Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimclsi (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem flimclsi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . . . 8 𝐽 = 𝐽
21flimfil 22866 . . . . . . 7 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
32ad2antlr 727 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimnei 22864 . . . . . . 7 ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
54adantll 714 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
6 simpll 767 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆𝐹)
7 filinn0 22757 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑦𝐹𝑆𝐹) → (𝑦𝑆) ≠ ∅)
83, 5, 6, 7syl3anc 1373 . . . . 5 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦𝑆) ≠ ∅)
98ralrimiva 3105 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅)
10 flimtop 22862 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
1110adantl 485 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12 filelss 22749 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑆𝐹) → 𝑆 𝐽)
1312ancoms 462 . . . . . 6 ((𝑆𝐹𝐹 ∈ (Fil‘ 𝐽)) → 𝑆 𝐽)
142, 13sylan2 596 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 𝐽)
151flimelbas 22865 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
1615adantl 485 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 𝐽)
171neindisj2 22020 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
1811, 14, 16, 17syl3anc 1373 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
199, 18mpbird 260 . . 3 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
2019ex 416 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
2120ssrdv 3907 1 (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2110  wne 2940  wral 3061  cin 3865  wss 3866  c0 4237  {csn 4541   cuni 4819  cfv 6380  (class class class)co 7213  Topctop 21790  clsccl 21915  neicnei 21994  Filcfil 22742   fLim cflim 22831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-fbas 20360  df-top 21791  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-fil 22743  df-flim 22836
This theorem is referenced by:  flimcls  22882  flimfcls  22923  cnextcn  22964  cmetss  24213  minveclem4  24329
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