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Theorem alexsubALTlem1 18083
Description: Lemma for alexsubALT 18087. 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 17463 . . 3  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 fitop 16978 . . . . 5  |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
32fveq2d 5735 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 ( fi `  J ) )  =  ( topGen `  J )
)
4 tgtop 17043 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
53, 4eqtr2d 2471 . . 3  |-  ( J  e.  Top  ->  J  =  ( topGen `  ( fi `  J ) ) )
61, 5syl 16 . 2  |-  ( J  e.  Comp  ->  J  =  ( topGen `  ( fi `  J ) ) )
7 vex 2961 . . . . 5  |-  c  e. 
_V
87elpw 3807 . . . 4  |-  ( c  e.  ~P J  <->  c  C_  J )
9 alexsubALT.1 . . . . . 6  |-  X  = 
U. J
109cmpcov 17457 . . . . 5  |-  ( ( J  e.  Comp  /\  c  C_  J  /\  X  = 
U. c )  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
)
11103exp 1153 . . . 4  |-  ( J  e.  Comp  ->  ( c 
C_  J  ->  ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
128, 11syl5bi 210 . . 3  |-  ( J  e.  Comp  ->  ( c  e.  ~P J  -> 
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
1312ralrimiv 2790 . 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 5731 . . . . . 6  |-  ( x  =  J  ->  ( fi `  x )  =  ( fi `  J
) )
1514fveq2d 5735 . . . . 5  |-  ( x  =  J  ->  ( topGen `
 ( fi `  x ) )  =  ( topGen `  ( fi `  J ) ) )
1615eqeq2d 2449 . . . 4  |-  ( x  =  J  ->  ( J  =  ( topGen `  ( fi `  x
) )  <->  J  =  ( topGen `  ( fi `  J ) ) ) )
17 pweq 3804 . . . . 5  |-  ( x  =  J  ->  ~P x  =  ~P J
)
1817raleqdv 2912 . . . 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 693 . . 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 3039 . 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 662 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 360   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457   Fincfn 7112   ficfi 7418   topGenctg 13670   Topctop 16963   Compccmp 17454
This theorem is referenced by:  alexsubALT  18087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-topgen 13672  df-top 16968  df-cmp 17455
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