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Theorem flimclsi 23926
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 2725 . . . . . . . 8 𝐽 = 𝐽
21flimfil 23917 . . . . . . 7 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
32ad2antlr 725 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimnei 23915 . . . . . . 7 ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
54adantll 712 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
6 simpll 765 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆𝐹)
7 filinn0 23808 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑦𝐹𝑆𝐹) → (𝑦𝑆) ≠ ∅)
83, 5, 6, 7syl3anc 1368 . . . . 5 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦𝑆) ≠ ∅)
98ralrimiva 3135 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅)
10 flimtop 23913 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
1110adantl 480 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12 filelss 23800 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑆𝐹) → 𝑆 𝐽)
1312ancoms 457 . . . . . 6 ((𝑆𝐹𝐹 ∈ (Fil‘ 𝐽)) → 𝑆 𝐽)
142, 13sylan2 591 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 𝐽)
151flimelbas 23916 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
1615adantl 480 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 𝐽)
171neindisj2 23071 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
1811, 14, 16, 17syl3anc 1368 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
199, 18mpbird 256 . . 3 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
2019ex 411 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
2120ssrdv 3982 1 (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wne 2929  wral 3050  cin 3943  wss 3944  c0 4322  {csn 4630   cuni 4909  cfv 6549  (class class class)co 7419  Topctop 22839  clsccl 22966  neicnei 23045  Filcfil 23793   fLim cflim 23882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-fbas 21293  df-top 22840  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-fil 23794  df-flim 23887
This theorem is referenced by:  flimcls  23933  flimfcls  23974  cnextcn  24015  cmetss  25288  minveclem4  25404
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