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Theorem flimclsi 22110
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 2799 . . . . . . . 8 𝐽 = 𝐽
21flimfil 22101 . . . . . . 7 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
32ad2antlr 719 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimnei 22099 . . . . . . 7 ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
54adantll 706 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
6 simpll 784 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆𝐹)
7 filinn0 21992 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑦𝐹𝑆𝐹) → (𝑦𝑆) ≠ ∅)
83, 5, 6, 7syl3anc 1491 . . . . 5 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦𝑆) ≠ ∅)
98ralrimiva 3147 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅)
10 flimtop 22097 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
1110adantl 474 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12 filelss 21984 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑆𝐹) → 𝑆 𝐽)
1312ancoms 451 . . . . . 6 ((𝑆𝐹𝐹 ∈ (Fil‘ 𝐽)) → 𝑆 𝐽)
142, 13sylan2 587 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 𝐽)
151flimelbas 22100 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
1615adantl 474 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 𝐽)
171neindisj2 21256 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
1811, 14, 16, 17syl3anc 1491 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
199, 18mpbird 249 . . 3 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
2019ex 402 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
2120ssrdv 3804 1 (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wne 2971  wral 3089  cin 3768  wss 3769  c0 4115  {csn 4368   cuni 4628  cfv 6101  (class class class)co 6878  Topctop 21026  clsccl 21151  neicnei 21230  Filcfil 21977   fLim cflim 22066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-fbas 20065  df-top 21027  df-cld 21152  df-ntr 21153  df-cls 21154  df-nei 21231  df-fil 21978  df-flim 22071
This theorem is referenced by:  flimcls  22117  flimfcls  22158  cnextcn  22199  cmetss  23442  minveclem4  23542
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