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

Theorem uffclsflim 21882
Description: The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
uffclsflim (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Proof of Theorem uffclsflim
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ufilfil 21755 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 fclsfnflim 21878 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
31, 2syl 17 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓))))
43biimpa 500 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∃𝑓 ∈ (Fil‘𝑋)(𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))
5 simprrr 822 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
6 simpll 805 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ (UFil‘𝑋))
7 simprl 809 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
8 simprrl 821 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹𝑓)
9 ufilmax 21758 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) → 𝐹 = 𝑓)
106, 7, 8, 9syl3anc 1366 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 = 𝑓)
1110oveq2d 6706 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽 fLim 𝐹) = (𝐽 fLim 𝑓))
125, 11eleqtrrd 2733 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝐹))
134, 12rexlimddv 3064 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
1413ex 449 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
1514ssrdv 3642 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
16 flimfcls 21877 . . 3 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
1716a1i 11 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹))
1815, 17eqssd 3653 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  wss 3607  cfv 5926  (class class class)co 6690  Filcfil 21696  UFilcufil 21750   fLim cflim 21785   fClus cfcls 21787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-fil 21697  df-ufil 21752  df-flim 21790  df-fcls 21792
This theorem is referenced by:  ufilcmp  21883  uffcfflf  21890
  Copyright terms: Public domain W3C validator