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Theorem nacsfg 26451
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 26450 . . . 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 451 . . 3  |-  ( C  e.  (NoeACS `  X
)  ->  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )
4 eqeq1 2394 . . . . 5  |-  ( s  =  S  ->  (
s  =  ( F `
 g )  <->  S  =  ( F `  g ) ) )
54rexbidv 2671 . . . 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 2994 . . 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 461 . 2  |-  ( ( S  e.  C  /\  C  e.  (NoeACS `  X
) )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
87ancoms 440 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 359    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651    i^i cin 3263   ~Pcpw 3743   ` cfv 5395   Fincfn 7046  mrClscmrc 13736  ACScacs 13738  NoeACScnacs 26448
This theorem is referenced by:  isnacs3  26456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-nacs 26449
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