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Theorem fclscmpi 18063
Description: Forward direction of fclscmp 18064. 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 17460 . . . 4  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 flimfnfcls.x . . . . . 6  |-  X  = 
U. J
32fclsval 18042 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) ,  (/) ) )
4 eqid 2438 . . . . . 6  |-  X  =  X
5 iftrue 3747 . . . . . 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 2486 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  |^|_ x  e.  F  ( ( cls `  J
) `  x )
)
81, 7sylan 459 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^|_ x  e.  F  ( ( cls `  J
) `  x )
)
9 fvex 5744 . . . 4  |-  ( ( cls `  J ) `
 x )  e. 
_V
109dfiin3 5127 . . 3  |-  |^|_ x  e.  F  ( ( cls `  J ) `  x )  =  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)
118, 10syl6eq 2486 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  = 
|^| ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )
12 simpl 445 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  J  e.  Comp )
1312adantr 453 . . . . . . 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 17886 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
1615adantll 696 . . . . . 6  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  X )
172clscld 17113 . . . . . 6  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
1814, 16, 17syl2anc 644 . . . . 5  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  ( Clsd `  J
) )
19 eqid 2438 . . . . 5  |-  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) )  =  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )
2018, 19fmptd 5895 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J ) )
21 frn 5599 . . . 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 449 . . . . . 6  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  F  e.  ( Fil `  X
) )
2423adantr 453 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  F  e.  ( Fil `  X ) )
25 simpr 449 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  e.  F )
262clsss3 17125 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( cls `  J
) `  x )  C_  X )
2714, 16, 26syl2anc 644 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  C_  X )
282sscls 17122 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  x  C_  X )  ->  x  C_  ( ( cls `  J ) `  x
) )
2914, 16, 28syl2anc 644 . . . . . . . . 9  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  x  C_  ( ( cls `  J ) `  x ) )
30 filss 17887 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  F )
3231, 19fmptd 5895 . . . . . . 7  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F )
33 frn 5599 . . . . . . 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 7431 . . . . . 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 644 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  ( fi `  F ) )
37 filfi 17893 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
3823, 37syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  F )  =  F )
3936, 38sseqtrd 3386 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  F )
40 0nelfil 17883 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
4123, 40syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  F )
4239, 41ssneldd 3353 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )
43 cmpfii 17474 . . 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 1185 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  =/=  (/) )
4511, 44eqnetrd 2621 1  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( J  fClus  F )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322   (/)c0 3630   ifcif 3741   U.cuni 4017   |^|cint 4052   |^|_ciin 4096    e. cmpt 4268   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083   ficfi 7417   Topctop 16960   Clsdccld 17082   clsccl 17084   Compccmp 17451   Filcfil 17879    fClus cfcls 17970
This theorem is referenced by:  fclscmp  18064  ufilcmp  18066  relcmpcmet  19271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-fbas 16701  df-top 16965  df-cld 17085  df-cls 17087  df-cmp 17452  df-fil 17880  df-fcls 17975
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