Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimclsi Structured version   Visualization version   GIF version

Theorem flimclsi 21592
 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 2610 . . . . . . . 8 𝐽 = 𝐽
21flimfil 21583 . . . . . . 7 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
32ad2antlr 759 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimnei 21581 . . . . . . 7 ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
54adantll 746 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦𝐹)
6 simpll 786 . . . . . 6 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆𝐹)
7 filinn0 21474 . . . . . 6 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑦𝐹𝑆𝐹) → (𝑦𝑆) ≠ ∅)
83, 5, 6, 7syl3anc 1318 . . . . 5 (((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦𝑆) ≠ ∅)
98ralrimiva 2949 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅)
10 flimtop 21579 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
1110adantl 481 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12 filelss 21466 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐽) ∧ 𝑆𝐹) → 𝑆 𝐽)
1312ancoms 468 . . . . . 6 ((𝑆𝐹𝐹 ∈ (Fil‘ 𝐽)) → 𝑆 𝐽)
142, 13sylan2 490 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 𝐽)
151flimelbas 21582 . . . . . 6 (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 𝐽)
1615adantl 481 . . . . 5 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 𝐽)
171neindisj2 20737 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 𝐽𝑥 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
1811, 14, 16, 17syl3anc 1318 . . . 4 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦𝑆) ≠ ∅))
199, 18mpbird 246 . . 3 ((𝑆𝐹𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
2019ex 449 . 2 (𝑆𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
2120ssrdv 3574 1 (𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549  Topctop 20517  clsccl 20632  neicnei 20711  Filcfil 21459   fLim cflim 21548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-fil 21460  df-flim 21553 This theorem is referenced by:  flimcls  21599  flimfcls  21640  cnextcn  21681  cmetss  22921  minveclem4  23011
 Copyright terms: Public domain W3C validator