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Theorem isfcls 23360
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 469 . 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 6875 . . . . . . . 8 (Fil‘𝑋) ⊆ ran Fil
32sseli 3940 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ran Fil)
4 filunibas 23232 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
54eqcomd 2742 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐹)
63, 5jca 512 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ran Fil ∧ 𝑋 = 𝐹))
7 filunirn 23233 . . . . . . 7 (𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))
8 fveq2 6842 . . . . . . . . 9 (𝑋 = 𝐹 → (Fil‘𝑋) = (Fil‘ 𝐹))
98eleq2d 2823 . . . . . . . 8 (𝑋 = 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐹)))
109biimparc 480 . . . . . . 7 ((𝐹 ∈ (Fil‘ 𝐹) ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
117, 10sylanb 581 . . . . . 6 ((𝐹 ran Fil ∧ 𝑋 = 𝐹) → 𝐹 ∈ (Fil‘𝑋))
126, 11impbii 208 . . . . 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 469 . . . 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 302 . 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 23292 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))
2019elmpocl 7595 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ran Fil))
21 fclsval.x . . . . . . 7 𝑋 = 𝐽
2221fclsval 23359 . . . . . 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 2823 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25 n0i 4293 . . . . . . 7 (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26 iffalse 4495 . . . . . . 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 563 . . . 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 4492 . . . . . . . 8 (𝑋 = 𝐹 → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3130adantl 482 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) = 𝑠𝐹 ((cls‘𝐽)‘𝑠))
3231eleq2d 2823 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 ∈ if(𝑋 = 𝐹, 𝑠𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠)))
33 elex 3463 . . . . . . . 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 23213 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘ 𝐹) → 𝐹 ≠ ∅)
367, 35sylbi 216 . . . . . . . . . 10 (𝐹 ran Fil → 𝐹 ≠ ∅)
3736ad2antlr 725 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → 𝐹 ≠ ∅)
38 r19.2z 4452 . . . . . . . . . 10 ((𝐹 ≠ ∅ ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
3938ex 413 . . . . . . . . 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 3463 . . . . . . . . 9 (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
4241rexlimivw 3148 . . . . . . . 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 4959 . . . . . . . 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 380 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ 𝑋 = 𝐹) → (𝐴 𝑠𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
4732, 46bitrd 278 . . . . 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 304 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
5020, 49biadanii 820 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil) ∧ (𝑋 = 𝐹 ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
511, 18, 503bitr4ri 303 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  c0 4282  ifcif 4486   cuni 4865   ciin 4955  ran crn 5634  cfv 6496  (class class class)co 7357  Topctop 22242  clsccl 22369  Filcfil 23196   fClus cfcls 23287
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 20793  df-fil 23197  df-fcls 23292
This theorem is referenced by:  fclsfil  23361  fclstop  23362  isfcls2  23364  fclssscls  23369  flimfcls  23377
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