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Theorem nacsfg 26780
Description: In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
nacsfg  |-  ( ( C  e.  (NoeACS `  X )  /\  S  e.  C )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
Distinct variable groups:    C, g    g, F    S, g    g, X

Proof of Theorem nacsfg
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . . . 5  |-  F  =  (mrCls `  C )
21isnacs 26779 . . . 4  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
32simprbi 450 . . 3  |-  ( C  e.  (NoeACS `  X
)  ->  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )
4 eqeq1 2289 . . . . 5  |-  ( s  =  S  ->  (
s  =  ( F `
 g )  <->  S  =  ( F `  g ) ) )
54rexbidv 2564 . . . 4  |-  ( s  =  S  ->  ( E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g )  <->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) ) )
65rspcva 2882 . . 3  |-  ( ( S  e.  C  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
73, 6sylan2 460 . 2  |-  ( ( S  e.  C  /\  C  e.  (NoeACS `  X
) )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
87ancoms 439 1  |-  ( ( C  e.  (NoeACS `  X )  /\  S  e.  C )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   ` cfv 5255   Fincfn 6863  mrClscmrc 13485  ACScacs 13487  NoeACScnacs 26777
This theorem is referenced by:  isnacs3  26785
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-nacs 26778
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