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Theorem isfcls 22014
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 684 . 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 6378 . . . . . . . 8 (Fil‘𝑋) ⊆ ran Fil
32sseli 3740 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ran Fil)
4 filunibas 21886 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
54eqcomd 2766 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐹)
63, 5jca 555 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ran Fil ∧ 𝑋 = 𝐹))
7 filunirn 21887 . . . . . . 7 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
8 fveq2 6352 . . . . . . . . 9 (𝑋 = 𝐹 → (Fil‘𝑋) = (Fil‘ 𝐹))
98eleq2d 2825 . . . . . . . 8 (𝑋 = 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐹)))
109biimparc 505 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
117, 10sylanb 490 . . . . . 6 ((𝐹 ran Fil ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
126, 11impbii 199 . . . . 5 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ran Fil ∧ 𝑋 = 𝐹))
1312anbi2i 732 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1413anbi1i 733 . . 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 1074 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
16 anass 684 . . . 4 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ran Fil ∧ 𝑋 = 𝐹)))
1716anbi1i 733 . . 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 292 . 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 21946 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))
2019elmpt2cl 7041 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil))
21 fclsval.x . . . . . . 7 𝑋 = 𝐽
2221fclsval 22013 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
237, 22sylan2b 493 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅))
2423eleq2d 2825 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25 n0i 4063 . . . . . . 7 (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26 iffalse 4239 . . . . . . 7 𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
2725, 26nsyl2 142 . . . . . 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 670 . . . 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 4236 . . . . . . . 8 (𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3130adantl 473 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3231eleq2d 2825 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠)))
33 elex 3352 . . . . . . . 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 21867 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ≠ ∅)
367, 35sylbi 207 . . . . . . . . . 10 (𝐹 ran Fil → 𝐹 ≠ ∅)
3736ad2antlr 765 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → 𝐹 ≠ ∅)
38 r19.2z 4204 . . . . . . . . . 10 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
3938ex 449 . . . . . . . . 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 3352 . . . . . . . . 9 (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
4241rexlimivw 3167 . . . . . . . 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 4677 . . . . . . . 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 368 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4732, 46bitrd 268 . . . . 5 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4847pm5.32da 676 . . . 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 294 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
5020, 49biadan2 677 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
511, 18, 503bitr4ri 293 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  c0 4058  ifcif 4230   cuni 4588   ciin 4673  ran crn 5267  cfv 6049  (class class class)co 6813  Topctop 20900  clsccl 21024  Filcfil 21850   fClus cfcls 21941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-fbas 19945  df-fil 21851  df-fcls 21946
This theorem is referenced by:  fclsfil  22015  fclstop  22016  isfcls2  22018  fclssscls  22023  flimfcls  22031
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