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Theorem cmpfiiin 26772
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 17122 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
31, 2syl 15 . . . 4  |-  ( ph  ->  J  e.  Top )
4 cmpfiiin.x . . . . 5  |-  X  = 
U. J
54topcld 16772 . . . 4  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
63, 5syl 15 . . 3  |-  ( ph  ->  X  e.  ( Clsd `  J ) )
7 cmpfiiin.s . . . . 5  |-  ( (
ph  /\  k  e.  I )  ->  S  e.  ( Clsd `  J
) )
84cldss 16766 . . . . 5  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
97, 8syl 15 . . . 4  |-  ( (
ph  /\  k  e.  I )  ->  S  C_  X )
109ralrimiva 2626 . . 3  |-  ( ph  ->  A. k  e.  I  S  C_  X )
11 riinint 4935 . . 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 642 . 2  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
136snssd 3760 . . . 4  |-  ( ph  ->  { X }  C_  ( Clsd `  J )
)
14 eqid 2283 . . . . . 6  |-  ( k  e.  I  |->  S )  =  ( k  e.  I  |->  S )
157, 14fmptd 5684 . . . . 5  |-  ( ph  ->  ( k  e.  I  |->  S ) : I --> ( Clsd `  J
) )
16 frn 5395 . . . . 5  |-  ( ( k  e.  I  |->  S ) : I --> ( Clsd `  J )  ->  ran  ( k  e.  I  |->  S )  C_  ( Clsd `  J ) )
1715, 16syl 15 . . . 4  |-  ( ph  ->  ran  ( k  e.  I  |->  S )  C_  ( Clsd `  J )
)
1813, 17unssd 3351 . . 3  |-  ( ph  ->  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) 
C_  ( Clsd `  J
) )
19 elin 3358 . . . . . . 7  |-  ( l  e.  ( ~P I  i^i  Fin )  <->  ( l  e.  ~P I  /\  l  e.  Fin ) )
20 elpwi 3633 . . . . . . . 8  |-  ( l  e.  ~P I  -> 
l  C_  I )
2120anim1i 551 . . . . . . 7  |-  ( ( l  e.  ~P I  /\  l  e.  Fin )  ->  ( l  C_  I  /\  l  e.  Fin ) )
2219, 21sylbi 187 . . . . . 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 2527 . . . . . . . 8  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  (/)  =/=  ( X  i^i  |^|_ k  e.  l  S ) )
25 df-ne 2448 . . . . . . . 8  |-  ( (/)  =/=  ( X  i^i  |^|_ k  e.  l  S
)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2624, 25bitri 240 . . . . . . 7  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2723, 26sylib 188 . . . . . 6  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2822, 27sylan2 460 . . . . 5  |-  ( (
ph  /\  l  e.  ( ~P I  i^i  Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2928nrexdv 2646 . . . 4  |-  ( ph  ->  -.  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
30 elrfirn2 26771 . . . . 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 642 . . . 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 292 . . 3  |-  ( ph  ->  -.  (/)  e.  ( fi
`  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) ) )
33 cmpfii 17136 . . 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 1182 . 2  |-  ( ph  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =/=  (/) )
3512, 34eqnetrd 2464 1  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   |^|cint 3862   |^|_ciin 3906    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255   Fincfn 6863   ficfi 7164   Topctop 16631   Clsdccld 16753   Compccmp 17113
This theorem is referenced by:  kelac1  27161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-top 16636  df-cld 16756  df-cmp 17114
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