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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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