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Theorem fclscmpi 20403
Description: Forward direction of fclscmp 20404. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
flimfnfcls.x  |-  X  = 
U. J
Assertion
Ref Expression
fclscmpi  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  =/=  (/) )

Proof of Theorem fclscmpi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cmptop 19768 . . . 4  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 flimfnfcls.x . . . . . 6  |-  X  = 
U. J
32fclsval 20382 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) ,  (/) ) )
4 eqid 2443 . . . . . 6  |-  X  =  X
54iftruei 3933 . . . . 5  |-  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J
) `  x ) ,  (/) )  =  |^|_ x  e.  F  ( ( cls `  J ) `
 x )
63, 5syl6eq 2500 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  |^|_ x  e.  F  ( ( cls `  J
) `  x )
)
71, 6sylan 471 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^|_ x  e.  F  ( ( cls `  J
) `  x )
)
8 fvex 5866 . . . 4  |-  ( ( cls `  J ) `
 x )  e. 
_V
98dfiin3 5248 . . 3  |-  |^|_ x  e.  F  ( ( cls `  J ) `  x )  =  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)
107, 9syl6eq 2500 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^| ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )
11 simpl 457 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Comp )
1211adantr 465 . . . . . . 7  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  J  e.  Comp )
1312, 1syl 16 . . . . . 6  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  J  e.  Top )
14 filelss 20226 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
1514adantll 713 . . . . . 6  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  X )
162clscld 19421 . . . . . 6  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
1713, 15, 16syl2anc 661 . . . . 5  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
18 eqid 2443 . . . . 5  |-  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) )  =  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )
1917, 18fmptd 6040 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J ) )
20 frn 5727 . . . 4  |-  ( ( x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  ( Clsd `  J ) )
2119, 20syl 16 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  ( Clsd `  J ) )
22 simpr 461 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  F  e.  ( Fil `  X
) )
2322adantr 465 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  F  e.  ( Fil `  X ) )
24 simpr 461 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  e.  F )
252clsss3 19433 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  C_  X )
2613, 15, 25syl2anc 661 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  C_  X )
272sscls 19430 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  ->  x  C_  ( ( cls `  J ) `  x
) )
2813, 15, 27syl2anc 661 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  ( ( cls `  J ) `  x ) )
29 filss 20227 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  ( ( cls `  J
) `  x )  C_  X  /\  x  C_  ( ( cls `  J
) `  x )
) )  ->  (
( cls `  J
) `  x )  e.  F )
3023, 24, 26, 28, 29syl13anc 1231 . . . . . . . 8  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  F )
3130, 18fmptd 6040 . . . . . . 7  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F )
32 frn 5727 . . . . . . 7  |-  ( ( x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  F )
3331, 32syl 16 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  F )
34 fiss 7886 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  F )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) )  C_  ( fi `  F ) )
3522, 33, 34syl2anc 661 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  ( fi `  F ) )
36 filfi 20233 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
3722, 36syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  F )  =  F )
3835, 37sseqtrd 3525 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  F )
39 0nelfil 20223 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
4022, 39syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  F )
4138, 40ssneldd 3492 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )
42 cmpfii 19782 . . 3  |-  ( ( J  e.  Comp  /\  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  ( Clsd `  J )  /\  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )  ->  |^| ran  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )  =/=  (/) )
4311, 21, 41, 42syl3anc 1229 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  =/=  (/) )
4410, 43eqnetrd 2736 1  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638    C_ wss 3461   (/)c0 3770   ifcif 3926   U.cuni 4234   |^|cint 4271   |^|_ciin 4316    |-> cmpt 4495   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   ficfi 7872   Topctop 19267   Clsdccld 19390   clsccl 19392   Compccmp 19759   Filcfil 20219    fClus cfcls 20310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-fbas 18290  df-top 19272  df-cld 19393  df-cls 19395  df-cmp 19760  df-fil 20220  df-fcls 20315
This theorem is referenced by:  fclscmp  20404  ufilcmp  20406  relcmpcmet  21628
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