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Theorem isfcls 22617
Description: A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
fclsval.x 𝑋 = 𝐽
Assertion
Ref Expression
isfcls (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Distinct variable groups:   𝐴,𝑠   𝐹,𝑠   𝑋,𝑠   𝐽,𝑠

Proof of Theorem isfcls
Dummy variables 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anass 472 . 2 ((((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
2 fvssunirn 6690 . . . . . . . 8 (Fil‘𝑋) ⊆ ran Fil
32sseli 3949 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ran Fil)
4 filunibas 22489 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
54eqcomd 2830 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐹)
63, 5jca 515 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ran Fil ∧ 𝑋 = 𝐹))
7 filunirn 22490 . . . . . . 7 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
8 fveq2 6661 . . . . . . . . 9 (𝑋 = 𝐹 → (Fil‘𝑋) = (Fil‘ 𝐹))
98eleq2d 2901 . . . . . . . 8 (𝑋 = 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐹)))
109biimparc 483 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
117, 10sylanb 584 . . . . . 6 ((𝐹 ran Fil ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
126, 11impbii 212 . . . . 5 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ran Fil ∧ 𝑋 = 𝐹))
1312anbi2i 625 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1413anbi1i 626 . . 3 (((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
15 df-3an 1086 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
16 anass 472 . . . 4 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1716anbi1i 626 . . 3 ((((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
1814, 15, 173bitr4i 306 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
19 df-fcls 22549 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))
2019elmpocl 7381 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil))
21 fclsval.x . . . . . . 7 𝑋 = 𝐽
2221fclsval 22616 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
237, 22sylan2b 596 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
2423eleq2d 2901 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25 n0i 4282 . . . . . . 7 (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26 iffalse 4459 . . . . . . 7 𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
2725, 26nsyl2 143 . . . . . 6 (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = 𝐹)
2827a1i 11 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = 𝐹))
2928pm4.71rd 566 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ (𝑋 = 𝐹𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))))
30 iftrue 4456 . . . . . . . 8 (𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3130adantl 485 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3231eleq2d 2901 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠)))
33 elex 3498 . . . . . . . 8 (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
3433a1i 11 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
35 filn0 22470 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ≠ ∅)
367, 35sylbi 220 . . . . . . . . . 10 (𝐹 ran Fil → 𝐹 ≠ ∅)
3736ad2antlr 726 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → 𝐹 ≠ ∅)
38 r19.2z 4423 . . . . . . . . . 10 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
3938ex 416 . . . . . . . . 9 (𝐹 ≠ ∅ → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4037, 39syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
41 elex 3498 . . . . . . . . 9 (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
4241rexlimivw 3274 . . . . . . . 8 (∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
4340, 42syl6 35 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
44 eliin 4910 . . . . . . . 8 (𝐴 ∈ V → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4544a1i 11 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ V → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
4634, 43, 45pm5.21ndd 384 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4732, 46bitrd 282 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4847pm5.32da 582 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → ((𝑋 = 𝐹𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅)) ↔ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
4924, 29, 483bitrd 308 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
5020, 49biadanii 821 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
511, 18, 503bitr4ri 307 1 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  Vcvv 3480  c0 4276  ifcif 4450   cuni 4824   ciin 4906  ran crn 5543  cfv 6343  (class class class)co 7149  Topctop 21501  clsccl 21626  Filcfil 22453   fClus cfcls 22544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-fbas 20542  df-fil 22454  df-fcls 22549
This theorem is referenced by:  fclsfil  22618  fclstop  22619  isfcls2  22621  fclssscls  22626  flimfcls  22634
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