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Theorem cmpfiiin 29168
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 19111 . . . . 5  |-  ( J  e.  Comp  ->  J  e. 
Top )
31, 2syl 16 . . . 4  |-  ( ph  ->  J  e.  Top )
4 cmpfiiin.x . . . . 5  |-  X  = 
U. J
54topcld 18752 . . . 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 18746 . . . . 5  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
97, 8syl 16 . . . 4  |-  ( (
ph  /\  k  e.  I )  ->  S  C_  X )
109ralrimiva 2820 . . 3  |-  ( ph  ->  A. k  e.  I  S  C_  X )
11 riinint 5191 . . 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 661 . 2  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
136snssd 4113 . . . 4  |-  ( ph  ->  { X }  C_  ( Clsd `  J )
)
14 eqid 2451 . . . . . 6  |-  ( k  e.  I  |->  S )  =  ( k  e.  I  |->  S )
157, 14fmptd 5963 . . . . 5  |-  ( ph  ->  ( k  e.  I  |->  S ) : I --> ( Clsd `  J
) )
16 frn 5660 . . . . 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 3627 . . 3  |-  ( ph  ->  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) 
C_  ( Clsd `  J
) )
19 elin 3634 . . . . . . 7  |-  ( l  e.  ( ~P I  i^i  Fin )  <->  ( l  e.  ~P I  /\  l  e.  Fin ) )
20 elpwi 3964 . . . . . . . 8  |-  ( l  e.  ~P I  -> 
l  C_  I )
2120anim1i 568 . . . . . . 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 necom 2715 . . . . . . . 8  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  (/)  =/=  ( X  i^i  |^|_ k  e.  l  S ) )
25 df-ne 2644 . . . . . . . 8  |-  ( (/)  =/=  ( X  i^i  |^|_ k  e.  l  S
)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2624, 25bitri 249 . . . . . . 7  |-  ( ( X  i^i  |^|_ k  e.  l  S )  =/=  (/)  <->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
2723, 26sylib 196 . . . . . 6  |-  ( (
ph  /\  ( l  C_  I  /\  l  e. 
Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2822, 27sylan2 474 . . . . 5  |-  ( (
ph  /\  l  e.  ( ~P I  i^i  Fin ) )  ->  -.  (/)  =  ( X  i^i  |^|_ k  e.  l  S ) )
2928nrexdv 2912 . . . 4  |-  ( ph  ->  -.  E. l  e.  ( ~P I  i^i 
Fin ) (/)  =  ( X  i^i  |^|_ k  e.  l  S )
)
30 elrfirn2 29167 . . . . 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 661 . . . 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 301 . . 3  |-  ( ph  ->  -.  (/)  e.  ( fi
`  ( { X }  u.  ran  ( k  e.  I  |->  S ) ) ) )
33 cmpfii 19125 . . 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 1219 . 2  |-  ( ph  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =/=  (/) )
3512, 34eqnetrd 2739 1  |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   E.wrex 2794    u. cun 3421    i^i cin 3422    C_ wss 3423   (/)c0 3732   ~Pcpw 3955   {csn 3972   U.cuni 4186   |^|cint 4223   |^|_ciin 4267    |-> cmpt 4445   ran crn 4936   -->wf 5509   ` cfv 5513   Fincfn 7407   ficfi 7758   Topctop 18611   Clsdccld 18733   Compccmp 19102
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-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-2o 7018  df-oadd 7021  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-fi 7759  df-top 18616  df-cld 18736  df-cmp 19103
This theorem is referenced by:  kelac1  29551
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