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Theorem alexsubALTlem1 20275
Description: Lemma for alexsubALT 20279. 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 19654 . . 3  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 fitop 19169 . . . . 5  |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
32fveq2d 5861 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 ( fi `  J ) )  =  ( topGen `  J )
)
4 tgtop 19234 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
53, 4eqtr2d 2502 . . 3  |-  ( J  e.  Top  ->  J  =  ( topGen `  ( fi `  J ) ) )
61, 5syl 16 . 2  |-  ( J  e.  Comp  ->  J  =  ( topGen `  ( fi `  J ) ) )
7 selpw 4010 . . . 4  |-  ( c  e.  ~P J  <->  c  C_  J )
8 alexsubALT.1 . . . . . 6  |-  X  = 
U. J
98cmpcov 19648 . . . . 5  |-  ( ( J  e.  Comp  /\  c  C_  J  /\  X  = 
U. c )  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
)
1093exp 1190 . . . 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 2869 . 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 5857 . . . . . 6  |-  ( x  =  J  ->  ( fi `  x )  =  ( fi `  J
) )
1413fveq2d 5861 . . . . 5  |-  ( x  =  J  ->  ( topGen `
 ( fi `  x ) )  =  ( topGen `  ( fi `  J ) ) )
1514eqeq2d 2474 . . . 4  |-  ( x  =  J  ->  ( J  =  ( topGen `  ( fi `  x
) )  <->  J  =  ( topGen `  ( fi `  J ) ) ) )
16 pweq 4006 . . . . 5  |-  ( x  =  J  ->  ~P x  =  ~P J
)
1716raleqdv 3057 . . . 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 3192 . 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 1374   E.wex 1591    e. wcel 1762   A.wral 2807   E.wrex 2808    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U.cuni 4238   ` cfv 5579   Fincfn 7506   ficfi 7859   topGenctg 14682   Topctop 19154   Compccmp 19645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-fin 7510  df-fi 7860  df-topgen 14688  df-top 19159  df-cmp 19646
This theorem is referenced by:  alexsubALT  20279
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