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

Theorem uffclsflim 22346
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 22219 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2 fclsfnflim 22342 . . . . . . 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 469 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∃𝑓 ∈ (Fil‘𝑋)(𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))
5 simprrr 769 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓))
6 simpll 754 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ (UFil‘𝑋))
7 simprl 758 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋))
8 simprrl 768 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹𝑓)
9 ufilmax 22222 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) → 𝐹 = 𝑓)
106, 7, 8, 9syl3anc 1351 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐹 = 𝑓)
1110oveq2d 6994 . . . . . 6 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽 fLim 𝐹) = (𝐽 fLim 𝑓))
125, 11eleqtrrd 2869 . . . . 5 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝐹𝑓𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝐹))
134, 12rexlimddv 3236 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
1413ex 405 . . 3 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
1514ssrdv 3866 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
16 flimfcls 22341 . . 3 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
1716a1i 11 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹))
1815, 17eqssd 3877 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3089  wss 3831  cfv 6190  (class class class)co 6978  Filcfil 22160  UFilcufil 22214   fLim cflim 22249   fClus cfcls 22251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-en 8309  df-fin 8312  df-fi 8672  df-fbas 20247  df-fg 20248  df-top 21209  df-topon 21226  df-cld 21334  df-ntr 21335  df-cls 21336  df-nei 21413  df-fil 22161  df-ufil 22216  df-flim 22254  df-fcls 22256
This theorem is referenced by:  ufilcmp  22347  uffcfflf  22354
  Copyright terms: Public domain W3C validator