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Theorem alexsubALTlem1 17954
Description: Lemma for alexsubALT 17958. 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 17339 . . 3  |-  ( J  e.  Comp  ->  J  e. 
Top )
2 fitop 16863 . . . . 5  |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
32fveq2d 5636 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 ( fi `  J ) )  =  ( topGen `  J )
)
4 tgtop 16928 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
53, 4eqtr2d 2399 . . 3  |-  ( J  e.  Top  ->  J  =  ( topGen `  ( fi `  J ) ) )
61, 5syl 15 . 2  |-  ( J  e.  Comp  ->  J  =  ( topGen `  ( fi `  J ) ) )
7 vex 2876 . . . . 5  |-  c  e. 
_V
87elpw 3720 . . . 4  |-  ( c  e.  ~P J  <->  c  C_  J )
9 alexsubALT.1 . . . . . 6  |-  X  = 
U. J
109cmpcov 17333 . . . . 5  |-  ( ( J  e.  Comp  /\  c  C_  J  /\  X  = 
U. c )  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d
)
11103exp 1151 . . . 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 2710 . 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 5632 . . . . . 6  |-  ( x  =  J  ->  ( fi `  x )  =  ( fi `  J
) )
1514fveq2d 5636 . . . . 5  |-  ( x  =  J  ->  ( topGen `
 ( fi `  x ) )  =  ( topGen `  ( fi `  J ) ) )
1615eqeq2d 2377 . . . 4  |-  ( x  =  J  ->  ( J  =  ( topGen `  ( fi `  x
) )  <->  J  =  ( topGen `  ( fi `  J ) ) ) )
17 pweq 3717 . . . . 5  |-  ( x  =  J  ->  ~P x  =  ~P J
)
1817raleqdv 2827 . . . 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 2954 . 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 1546    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   U.cuni 3929   ` cfv 5358   Fincfn 7006   ficfi 7311   topGenctg 13552   Topctop 16848   Compccmp 17330
This theorem is referenced by:  alexsubALT  17958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-fin 7010  df-fi 7312  df-topgen 13554  df-top 16853  df-cmp 17331
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