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Theorem fclscmpi 19718
Description: Forward direction of fclscmp 19719. 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 19114 . . . 4  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 flimfnfcls.x . . . . . 6  |-  X  = 
U. J
32fclsval 19697 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  X ) )  -> 
( J  fClus  F )  =  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J ) `
 x ) ,  (/) ) )
4 eqid 2451 . . . . . 6  |-  X  =  X
54iftruei 3896 . . . . 5  |-  if ( X  =  X ,  |^|_ x  e.  F  ( ( cls `  J
) `  x ) ,  (/) )  =  |^|_ x  e.  F  ( ( cls `  J ) `
 x )
63, 5syl6eq 2508 . . . 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 5799 . . . 4  |-  ( ( cls `  J ) `
 x )  e. 
_V
98dfiin3 5193 . . 3  |-  |^|_ x  e.  F  ( ( cls `  J ) `  x )  =  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)
107, 9syl6eq 2508 . 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 19541 . . . . . . 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 18767 . . . . . 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 2451 . . . . 5  |-  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) )  =  ( x  e.  F  |->  ( ( cls `  J ) `  x
) )
1917, 18fmptd 5966 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> ( Clsd `  J ) )
20 frn 5663 . . . 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 18779 . . . . . . . . . 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 18776 . . . . . . . . . 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 19542 . . . . . . . . 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 1221 . . . . . . . 8  |-  ( ( ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  /\  x  e.  F )  ->  ( ( cls `  J
) `  x )  e.  F )
3130, 18fmptd 5966 . . . . . . 7  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  F  |->  ( ( cls `  J
) `  x )
) : F --> F )
32 frn 5663 . . . . . . 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 7775 . . . . . 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 19548 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( fi `  F )  =  F )
3722, 36syl 16 . . . . 5  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  F )  =  F )
3835, 37sseqtrd 3490 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J ) `
 x ) ) )  C_  F )
39 0nelfil 19538 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
4022, 39syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  F )
4138, 40ssneldd 3457 . . 3  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  -.  (/) 
e.  ( fi `  ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
) ) )
42 cmpfii 19128 . . 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 1219 . 2  |-  ( ( J  e.  Comp  /\  F  e.  ( Fil `  X
) )  ->  |^| ran  ( x  e.  F  |->  ( ( cls `  J
) `  x )
)  =/=  (/) )
4410, 43eqnetrd 2741 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 1370    e. wcel 1758    =/= wne 2644    C_ wss 3426   (/)c0 3735   ifcif 3889   U.cuni 4189   |^|cint 4226   |^|_ciin 4270    |-> cmpt 4448   ran crn 4939   -->wf 5512   ` cfv 5516  (class class class)co 6190   ficfi 7761   Topctop 18614   Clsdccld 18736   clsccl 18738   Compccmp 19105   Filcfil 19534    fClus cfcls 19625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fi 7762  df-fbas 17923  df-top 18619  df-cld 18739  df-cls 18741  df-cmp 19106  df-fil 19535  df-fcls 19630
This theorem is referenced by:  fclscmp  19719  ufilcmp  19721  relcmpcmet  20943
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