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Theorem fclscmpi 17975
Description: Forward direction of fclscmp 17976. 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 17373 . . . 4  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 flimfnfcls.x . . . . . 6  |-  X  = 
U. J
32fclsval 17954 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) ,  (/) ) )
4 eqid 2380 . . . . . 6  |-  X  =  X
5 iftrue 3681 . . . . . 6  |-  ( X  =  X  ->  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J
) `  x ) ,  (/) )  =  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) )
64, 5ax-mp 8 . . . . 5  |-  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J
) `  x ) ,  (/) )  =  |^|_ x  e.  F  ( ( cls `  J ) `
 x )
73, 6syl6eq 2428 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  |^|_ x  e.  F  ( ( cls `  J
) `  x )
)
81, 7sylan 458 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^|_ x  e.  F  ( ( cls `  J
) `  x )
)
9 fvex 5675 . . . 4  |-  ( ( cls `  J ) `
 x )  e. 
_V
109dfiin3 5058 . . 3  |-  |^|_ x  e.  F  ( ( cls `  J ) `  x )  =  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)
118, 10syl6eq 2428 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^| ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )
12 simpl 444 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Comp )
1312adantr 452 . . . . . . 7  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  J  e.  Comp )
1413, 1syl 16 . . . . . 6  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  J  e.  Top )
15 filelss 17798 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
1615adantll 695 . . . . . 6  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  X )
172clscld 17027 . . . . . 6  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
1814, 16, 17syl2anc 643 . . . . 5  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
19 eqid 2380 . . . . 5  |-  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) )  =  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )
2018, 19fmptd 5825 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J ) )
21 frn 5530 . . . 4  |-  ( ( x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  ( Clsd `  J ) )
2220, 21syl 16 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  ( Clsd `  J ) )
23 simpr 448 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  F  e.  ( Fil `  X
) )
2423adantr 452 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  F  e.  ( Fil `  X ) )
25 simpr 448 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  e.  F )
262clsss3 17039 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  C_  X )
2714, 16, 26syl2anc 643 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  C_  X )
282sscls 17036 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  ->  x  C_  ( ( cls `  J ) `  x
) )
2914, 16, 28syl2anc 643 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  ( ( cls `  J ) `  x ) )
30 filss 17799 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  ( ( cls `  J
) `  x )  C_  X  /\  x  C_  ( ( cls `  J
) `  x )
) )  ->  (
( cls `  J
) `  x )  e.  F )
3124, 25, 27, 29, 30syl13anc 1186 . . . . . . . 8  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  F )
3231, 19fmptd 5825 . . . . . . 7  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F )
33 frn 5530 . . . . . . 7  |-  ( ( x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  F )
3432, 33syl 16 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  C_  F )
35 fiss 7357 . . . . . 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 ) )
3623, 34, 35syl2anc 643 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  ( fi `  F ) )
37 filfi 17805 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
3823, 37syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  F )  =  F )
3936, 38sseqtrd 3320 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  F )
40 0nelfil 17795 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
4123, 40syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  F )
4239, 41ssneldd 3287 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )
43 cmpfii 17387 . . 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
) )  =/=  (/) )
4412, 22, 42, 43syl3anc 1184 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  =/=  (/) )
4511, 44eqnetrd 2561 1  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543    C_ wss 3256   (/)c0 3564   ifcif 3675   U.cuni 3950   |^|cint 3985   |^|_ciin 4029    e. cmpt 4200   ran crn 4812   -->wf 5383   ` cfv 5387  (class class class)co 6013   ficfi 7343   Topctop 16874   Clsdccld 16996   clsccl 16998   Compccmp 17364   Filcfil 17791    fClus cfcls 17882
This theorem is referenced by:  fclscmp  17976  ufilcmp  17978  relcmpcmet  19133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-fbas 16616  df-top 16879  df-cld 16999  df-cls 17001  df-cmp 17365  df-fil 17792  df-fcls 17887
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