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Theorem fclscmpi 18022
Description: Forward direction of fclscmp 18023. 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 17420 . . . 4  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 flimfnfcls.x . . . . . 6  |-  X  = 
U. J
32fclsval 18001 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) ,  (/) ) )
4 eqid 2412 . . . . . 6  |-  X  =  X
5 iftrue 3713 . . . . . 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 2460 . . . 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 5709 . . . 4  |-  ( ( cls `  J ) `
 x )  e. 
_V
109dfiin3 5092 . . 3  |-  |^|_ x  e.  F  ( ( cls `  J ) `  x )  =  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)
118, 10syl6eq 2460 . 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 17845 . . . . . . 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 17074 . . . . . 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 2412 . . . . 5  |-  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) )  =  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )
2018, 19fmptd 5860 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J ) )
21 frn 5564 . . . 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 17086 . . . . . . . . . 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 17083 . . . . . . . . . 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 17846 . . . . . . . . 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 5860 . . . . . . 7  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F )
33 frn 5564 . . . . . . 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 7395 . . . . . 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 17852 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
3823, 37syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  F )  =  F )
3936, 38sseqtrd 3352 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  F )
40 0nelfil 17842 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
4123, 40syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  F )
4239, 41ssneldd 3319 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )
43 cmpfii 17434 . . 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 2593 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 1721    =/= wne 2575    C_ wss 3288   (/)c0 3596   ifcif 3707   U.cuni 3983   |^|cint 4018   |^|_ciin 4062    e. cmpt 4234   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   ficfi 7381   Topctop 16921   Clsdccld 17043   clsccl 17045   Compccmp 17411   Filcfil 17838    fClus cfcls 17929
This theorem is referenced by:  fclscmp  18023  ufilcmp  18025  relcmpcmet  19230
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-fbas 16662  df-top 16926  df-cld 17046  df-cls 17048  df-cmp 17412  df-fil 17839  df-fcls 17934
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