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Theorem nacsfg 26750
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 26749 . . . 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 2441 . . . . 5  |-  ( s  =  S  ->  (
s  =  ( F `
 g )  <->  S  =  ( F `  g ) ) )
54rexbidv 2718 . . . 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 3042 . . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311   ~Pcpw 3791   ` cfv 5446   Fincfn 7101  mrClscmrc 13800  ACScacs 13802  NoeACScnacs 26747
This theorem is referenced by:  isnacs3  26755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-nacs 26748
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