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Theorem cmpfiiin 26705
Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypotheses
Ref Expression
cmpfiiin.x  |-  X  = 
U. J
cmpfiiin.j  |-  ( ph  ->  J  e.  Comp )
cmpfiiin.s  |-  ( (
ph  /\  k  e.  I )  ->  S  e.  ( Clsd `  J
) )
cmpfiiin.z  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  -> 
( X  i^i  |^|_ k  e.  l  S
)  =/=  (/) )
Assertion
Ref Expression
cmpfiiin  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =/=  (/) )
Distinct variable groups:    ph, k, l   
k, I, l    k, J, l    S, l    k, X, l
Allowed substitution hint:    S( k)

Proof of Theorem cmpfiiin
StepHypRef Expression
1 cmpfiiin.j . . . . 5  |-  ( ph  ->  J  e.  Comp )
2 cmptop 17448 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
31, 2syl 16 . . . 4  |-  ( ph  ->  J  e.  Top )
4 cmpfiiin.x . . . . 5  |-  X  = 
U. J
54topcld 17089 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
63, 5syl 16 . . 3  |-  ( ph  ->  X  e.  ( Clsd `  J ) )
7 cmpfiiin.s . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  S  e.  ( Clsd `  J
) )
84cldss 17083 . . . . 5  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
97, 8syl 16 . . . 4  |-  ( (
ph  /\  k  e.  I )  ->  S  C_  X )
109ralrimiva 2781 . . 3  |-  ( ph  ->  A. k  e.  I  S  C_  X )
11 riinint 5118 . . 3  |-  ( ( X  e.  ( Clsd `  J )  /\  A. k  e.  I  S  C_  X )  ->  ( X  i^i  |^|_ k  e.  I  S )  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
126, 10, 11syl2anc 643 . 2  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
136snssd 3935 . . . 4  |-  ( ph  ->  { X }  C_  ( Clsd `  J )
)
14 eqid 2435 . . . . . 6  |-  ( k  e.  I  |->  S )  =  ( k  e.  I  |->  S )
157, 14fmptd 5885 . . . . 5  |-  ( ph  ->  ( k  e.  I  |->  S ) : I --> ( Clsd `  J
) )
16 frn 5589 . . . . 5  |-  ( ( k  e.  I  |->  S ) : I --> ( Clsd `  J )  ->  ran  ( k  e.  I  |->  S )  C_  ( Clsd `  J ) )
1715, 16syl 16 . . . 4  |-  ( ph  ->  ran  ( k  e.  I  |->  S )  C_  ( Clsd `  J )
)
1813, 17unssd 3515 . . 3  |-  ( ph  ->  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) 
C_  ( Clsd `  J
) )
19 elin 3522 . . . . . . 7  |-  ( l  e.  ( ~P I  i^i  Fin )  <->  ( l  e.  ~P I  /\  l  e.  Fin ) )
20 elpwi 3799 . . . . . . . 8  |-  ( l  e.  ~P I  -> 
l  C_  I )
2120anim1i 552 . . . . . . 7  |-  ( ( l  e.  ~P I  /\  l  e.  Fin )  ->  ( l  C_  I  /\  l  e.  Fin ) )
2219, 21sylbi 188 . . . . . 6  |-  ( l  e.  ( ~P I  i^i  Fin )  ->  (
l  C_  I  /\  l  e.  Fin )
)
23 cmpfiiin.z . . . . . . 7  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  -> 
( X  i^i  |^|_ k  e.  l  S
)  =/=  (/) )
24 necom 2679 . . . . . . . 8  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  (/)  =/=  ( X  i^i  |^|_ k  e.  l  S ) )
25 df-ne 2600 . . . . . . . 8  |-  ( (/)  =/=  ( X  i^i  |^|_ k  e.  l  S
)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2624, 25bitri 241 . . . . . . 7  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2723, 26sylib 189 . . . . . 6  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2822, 27sylan2 461 . . . . 5  |-  ( (
ph  /\  l  e.  ( ~P I  i^i  Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2928nrexdv 2801 . . . 4  |-  ( ph  ->  -.  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
30 elrfirn2 26704 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  A. k  e.  I  S  C_  X )  ->  ( (/) 
e.  ( fi `  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )  <->  E. l  e.  ( ~P I  i^i  Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S ) ) )
316, 10, 30syl2anc 643 . . . 4  |-  ( ph  ->  ( (/)  e.  ( fi `  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )  <->  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
) )
3229, 31mtbird 293 . . 3  |-  ( ph  ->  -.  (/)  e.  ( fi
`  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) ) )
33 cmpfii 17462 . . 3  |-  ( ( J  e.  Comp  /\  ( { X }  u.  ran  ( k  e.  I  |->  S ) )  C_  ( Clsd `  J )  /\  -.  (/)  e.  ( fi
`  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) ) )  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =/=  (/) )
341, 18, 32, 33syl3anc 1184 . 2  |-  ( ph  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =/=  (/) )
3512, 34eqnetrd 2616 1  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007   |^|cint 4042   |^|_ciin 4086    e. cmpt 4258   ran crn 4871   -->wf 5442   ` cfv 5446   Fincfn 7101   ficfi 7407   Topctop 16948   Clsdccld 17070   Compccmp 17439
This theorem is referenced by:  kelac1  27093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-top 16953  df-cld 17073  df-cmp 17440
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