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Theorem cmpfiiin 30869
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 20062 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
31, 2syl 16 . . . 4  |-  ( ph  ->  J  e.  Top )
4 cmpfiiin.x . . . . 5  |-  X  = 
U. J
54topcld 19703 . . . 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 19697 . . . . 5  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
97, 8syl 16 . . . 4  |-  ( (
ph  /\  k  e.  I )  ->  S  C_  X )
109ralrimiva 2868 . . 3  |-  ( ph  ->  A. k  e.  I  S  C_  X )
11 riinint 5248 . . 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 659 . 2  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
136snssd 4161 . . . 4  |-  ( ph  ->  { X }  C_  ( Clsd `  J )
)
14 eqid 2454 . . . . . 6  |-  ( k  e.  I  |->  S )  =  ( k  e.  I  |->  S )
157, 14fmptd 6031 . . . . 5  |-  ( ph  ->  ( k  e.  I  |->  S ) : I --> ( Clsd `  J
) )
16 frn 5719 . . . . 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 3666 . . 3  |-  ( ph  ->  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) 
C_  ( Clsd `  J
) )
19 elin 3673 . . . . . . 7  |-  ( l  e.  ( ~P I  i^i  Fin )  <->  ( l  e.  ~P I  /\  l  e.  Fin ) )
20 elpwi 4008 . . . . . . . 8  |-  ( l  e.  ~P I  -> 
l  C_  I )
2120anim1i 566 . . . . . . 7  |-  ( ( l  e.  ~P I  /\  l  e.  Fin )  ->  ( l  C_  I  /\  l  e.  Fin ) )
2219, 21sylbi 195 . . . . . 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 nesym 2726 . . . . . . 7  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2523, 24sylib 196 . . . . . 6  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2622, 25sylan2 472 . . . . 5  |-  ( (
ph  /\  l  e.  ( ~P I  i^i  Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2726nrexdv 2910 . . . 4  |-  ( ph  ->  -.  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
28 elrfirn2 30868 . . . . 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 ) ) )
296, 10, 28syl2anc 659 . . . 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 )
) )
3027, 29mtbird 299 . . 3  |-  ( ph  ->  -.  (/)  e.  ( fi
`  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) ) )
31 cmpfii 20076 . . 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 ) )  =/=  (/) )
321, 18, 30, 31syl3anc 1226 . 2  |-  ( ph  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =/=  (/) )
3312, 32eqnetrd 2747 1  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   |^|cint 4271   |^|_ciin 4316    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570   Fincfn 7509   ficfi 7862   Topctop 19561   Clsdccld 19684   Compccmp 20053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-top 19566  df-cld 19687  df-cmp 20054
This theorem is referenced by:  kelac1  31248
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