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Theorem flimclsi 23986
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 23977 . . . . . . 7 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
32ad2antlr 727 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimnei 23975 . . . . . . 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 23868 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑦𝐹𝑆𝐹) → (𝑦𝑆) ≠ ∅)
83, 5, 6, 7syl3anc 1373 . . . . 5 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦𝑆) ≠ ∅)
98ralrimiva 3146 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅)
10 flimtop 23973 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
1110adantl 481 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12 filelss 23860 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑆𝐹) → 𝑆 𝐽)
1312ancoms 458 . . . . . 6 ((𝑆𝐹𝐹 ∈ (Fil‘ 𝐽)) → 𝑆 𝐽)
142, 13sylan2 593 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 𝐽)
151flimelbas 23976 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
1615adantl 481 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 𝐽)
171neindisj2 23131 . . . . 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 257 . . 3 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
2019ex 412 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
2120ssrdv 3989 1 (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wne 2940  wral 3061  cin 3950  wss 3951  c0 4333  {csn 4626   cuni 4907  cfv 6561  (class class class)co 7431  Topctop 22899  clsccl 23026  neicnei 23105  Filcfil 23853   fLim cflim 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-fil 23854  df-flim 23947
This theorem is referenced by:  flimcls  23993  flimfcls  24034  cnextcn  24075  cmetss  25350  minveclem4  25466
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