Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nacsfg Structured version   Unicode version

Theorem nacsfg 34963
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 34962 . . . 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 462 . . 3  |-  ( C  e.  (NoeACS `  X
)  ->  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )
4 eqeq1 2404 . . . . 5  |-  ( s  =  S  ->  (
s  =  ( F `
 g )  <->  S  =  ( F `  g ) ) )
54rexbidv 2915 . . . 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 3155 . . 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 472 . 2  |-  ( ( S  e.  C  /\  C  e.  (NoeACS `  X
) )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
87ancoms 451 1  |-  ( ( C  e.  (NoeACS `  X )  /\  S  e.  C )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752    i^i cin 3410   ~Pcpw 3952   ` cfv 5523   Fincfn 7472  mrClscmrc 15087  ACScacs 15089  NoeACScnacs 34960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-nacs 34961
This theorem is referenced by:  isnacs3  34968
  Copyright terms: Public domain W3C validator