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Theorem alexsubALTlem1 20524
Description: Lemma for alexsubALT 20528. 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 19872 . . 3  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 fitop 19386 . . . . 5  |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
32fveq2d 5860 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 ( fi `  J ) )  =  ( topGen `  J )
)
4 tgtop 19452 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
53, 4eqtr2d 2485 . . 3  |-  ( J  e.  Top  ->  J  =  ( topGen `  ( fi `  J ) ) )
61, 5syl 16 . 2  |-  ( J  e.  Comp  ->  J  =  ( topGen `  ( fi `  J ) ) )
7 selpw 4004 . . . 4  |-  ( c  e.  ~P J  <->  c  C_  J )
8 alexsubALT.1 . . . . . 6  |-  X  = 
U. J
98cmpcov 19866 . . . . 5  |-  ( ( J  e.  Comp  /\  c  C_  J  /\  X  = 
U. c )  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
)
1093exp 1196 . . . 4  |-  ( J  e.  Comp  ->  ( c 
C_  J  ->  ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
117, 10syl5bi 217 . . 3  |-  ( J  e.  Comp  ->  ( c  e.  ~P J  -> 
( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
1211ralrimiv 2855 . 2  |-  ( J  e.  Comp  ->  A. c  e.  ~P  J ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
) )
13 fveq2 5856 . . . . . 6  |-  ( x  =  J  ->  ( fi `  x )  =  ( fi `  J
) )
1413fveq2d 5860 . . . . 5  |-  ( x  =  J  ->  ( topGen `
 ( fi `  x ) )  =  ( topGen `  ( fi `  J ) ) )
1514eqeq2d 2457 . . . 4  |-  ( x  =  J  ->  ( J  =  ( topGen `  ( fi `  x
) )  <->  J  =  ( topGen `  ( fi `  J ) ) ) )
16 pweq 4000 . . . . 5  |-  ( x  =  J  ->  ~P x  =  ~P J
)
1716raleqdv 3046 . . . 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
) ) )
1815, 17anbi12d 710 . . 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
) ) ) )
1918spcegv 3181 . 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 ) ) ) )
206, 12, 19mp2and 679 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   A.wral 2793   E.wrex 2794    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   U.cuni 4234   ` cfv 5578   Fincfn 7518   ficfi 7872   topGenctg 14816   Topctop 19371   Compccmp 19863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-fin 7522  df-fi 7873  df-topgen 14822  df-top 19376  df-cmp 19864
This theorem is referenced by:  alexsubALT  20528
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