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Theorem cmpfiiin 25920
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 17178 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
31, 2syl 15 . . . 4  |-  ( ph  ->  J  e.  Top )
4 cmpfiiin.x . . . . 5  |-  X  = 
U. J
54topcld 16828 . . . 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 16822 . . . . 5  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
97, 8syl 15 . . . 4  |-  ( (
ph  /\  k  e.  I )  ->  S  C_  X )
109ralrimiva 2660 . . 3  |-  ( ph  ->  A. k  e.  I  S  C_  X )
11 riinint 4972 . . 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 3797 . . . 4  |-  ( ph  ->  { X }  C_  ( Clsd `  J )
)
14 eqid 2316 . . . . . 6  |-  ( k  e.  I  |->  S )  =  ( k  e.  I  |->  S )
157, 14fmptd 5722 . . . . 5  |-  ( ph  ->  ( k  e.  I  |->  S ) : I --> ( Clsd `  J
) )
16 frn 5433 . . . . 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 3385 . . 3  |-  ( ph  ->  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) 
C_  ( Clsd `  J
) )
19 elin 3392 . . . . . . 7  |-  ( l  e.  ( ~P I  i^i  Fin )  <->  ( l  e.  ~P I  /\  l  e.  Fin ) )
20 elpwi 3667 . . . . . . . 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 2560 . . . . . . . 8  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  (/)  =/=  ( X  i^i  |^|_ k  e.  l  S ) )
25 df-ne 2481 . . . . . . . 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 2680 . . . 4  |-  ( ph  ->  -.  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
30 elrfirn2 25919 . . . . 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 17192 . . 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 2497 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578    u. cun 3184    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   {csn 3674   U.cuni 3864   |^|cint 3899   |^|_ciin 3943    e. cmpt 4114   ran crn 4727   -->wf 5288   ` cfv 5292   Fincfn 6906   ficfi 7209   Topctop 16687   Clsdccld 16809   Compccmp 17169
This theorem is referenced by:  kelac1  26309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-top 16692  df-cld 16812  df-cmp 17170
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