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Theorem alexsubALTlem1 17741
Description: Lemma for alexsubALT 17745. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
Hypothesis
Ref Expression
alexsubALT.1  |-  X  = 
U. J
Assertion
Ref Expression
alexsubALTlem1  |-  ( J  e.  Comp  ->  E. x
( J  =  (
topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
Distinct variable groups:    c, d, x, J    X, c, d, x

Proof of Theorem alexsubALTlem1
StepHypRef Expression
1 cmptop 17122 . . 3  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 fitop 16646 . . . . 5  |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
32fveq2d 5529 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 ( fi `  J ) )  =  ( topGen `  J )
)
4 tgtop 16711 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
53, 4eqtr2d 2316 . . 3  |-  ( J  e.  Top  ->  J  =  ( topGen `  ( fi `  J ) ) )
61, 5syl 15 . 2  |-  ( J  e.  Comp  ->  J  =  ( topGen `  ( fi `  J ) ) )
7 vex 2791 . . . . 5  |-  c  e. 
_V
87elpw 3631 . . . 4  |-  ( c  e.  ~P J  <->  c  C_  J )
9 alexsubALT.1 . . . . . 6  |-  X  = 
U. J
109cmpcov 17116 . . . . 5  |-  ( ( J  e.  Comp  /\  c  C_  J  /\  X  = 
U. c )  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
)
11103exp 1150 . . . 4  |-  ( J  e.  Comp  ->  ( c 
C_  J  ->  ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
128, 11syl5bi 208 . . 3  |-  ( J  e.  Comp  ->  ( c  e.  ~P J  -> 
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
1312ralrimiv 2625 . 2  |-  ( J  e.  Comp  ->  A. c  e.  ~P  J ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
) )
14 fveq2 5525 . . . . . 6  |-  ( x  =  J  ->  ( fi `  x )  =  ( fi `  J
) )
1514fveq2d 5529 . . . . 5  |-  ( x  =  J  ->  ( topGen `
 ( fi `  x ) )  =  ( topGen `  ( fi `  J ) ) )
1615eqeq2d 2294 . . . 4  |-  ( x  =  J  ->  ( J  =  ( topGen `  ( fi `  x
) )  <->  J  =  ( topGen `  ( fi `  J ) ) ) )
17 pweq 3628 . . . . 5  |-  ( x  =  J  ->  ~P x  =  ~P J
)
1817raleqdv 2742 . . . 4  |-  ( x  =  J  ->  ( A. c  e.  ~P  x ( X  = 
U. c  ->  E. d  e.  ( ~P c  i^i 
Fin ) X  = 
U. d )  <->  A. c  e.  ~P  J ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
) ) )
1916, 18anbi12d 691 . . 3  |-  ( x  =  J  ->  (
( J  =  (
topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) )  <->  ( J  =  ( topGen `  ( fi `  J ) )  /\  A. c  e. 
~P  J ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
) ) ) )
2019spcegv 2869 . 2  |-  ( J  e.  Comp  ->  ( ( J  =  ( topGen `  ( fi `  J
) )  /\  A. c  e.  ~P  J
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) )  ->  E. x ( J  =  ( topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) ) )
216, 13, 20mp2and 660 1  |-  ( J  e.  Comp  ->  E. x
( J  =  (
topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   Fincfn 6863   ficfi 7164   topGenctg 13342   Topctop 16631   Compccmp 17113
This theorem is referenced by:  alexsubALT  17745
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-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-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-topgen 13344  df-top 16636  df-cmp 17114
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