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Theorem isfcls 24017
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 468 . 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 6939 . . . . . . . 8 (Fil‘𝑋) ⊆ ran Fil
32sseli 3979 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ran Fil)
4 filunibas 23889 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
54eqcomd 2743 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐹)
63, 5jca 511 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ran Fil ∧ 𝑋 = 𝐹))
7 filunirn 23890 . . . . . . 7 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
8 fveq2 6906 . . . . . . . . 9 (𝑋 = 𝐹 → (Fil‘𝑋) = (Fil‘ 𝐹))
98eleq2d 2827 . . . . . . . 8 (𝑋 = 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐹)))
109biimparc 479 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
117, 10sylanb 581 . . . . . 6 ((𝐹 ran Fil ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
126, 11impbii 209 . . . . 5 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ran Fil ∧ 𝑋 = 𝐹))
1312anbi2i 623 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1413anbi1i 624 . . 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 1089 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
16 anass 468 . . . 4 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1716anbi1i 624 . . 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 303 . 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 23949 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))
2019elmpocl 7674 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil))
21 fclsval.x . . . . . . 7 𝑋 = 𝐽
2221fclsval 24016 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
237, 22sylan2b 594 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
2423eleq2d 2827 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25 n0i 4340 . . . . . . 7 (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26 iffalse 4534 . . . . . . 7 𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
2725, 26nsyl2 141 . . . . . 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 562 . . . 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 4531 . . . . . . . 8 (𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3130adantl 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3231eleq2d 2827 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠)))
33 elex 3501 . . . . . . . 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 23870 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ≠ ∅)
367, 35sylbi 217 . . . . . . . . . 10 (𝐹 ran Fil → 𝐹 ≠ ∅)
3736ad2antlr 727 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → 𝐹 ≠ ∅)
38 r19.2z 4495 . . . . . . . . . 10 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
3938ex 412 . . . . . . . . 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 3501 . . . . . . . . 9 (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
4241rexlimivw 3151 . . . . . . . 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 4996 . . . . . . . 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 379 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4732, 46bitrd 279 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4847pm5.32da 579 . . . 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 305 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
5020, 49biadanii 822 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
511, 18, 503bitr4ri 304 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  c0 4333  ifcif 4525   cuni 4907   ciin 4992  ran crn 5686  cfv 6561  (class class class)co 7431  Topctop 22899  clsccl 23026  Filcfil 23853   fClus cfcls 23944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fil 23854  df-fcls 23949
This theorem is referenced by:  fclsfil  24018  fclstop  24019  isfcls2  24021  fclssscls  24026  flimfcls  24034
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