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 29209
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 29208 . . . 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 464 . . 3  |-  ( C  e.  (NoeACS `  X
)  ->  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )
4 eqeq1 2458 . . . . 5  |-  ( s  =  S  ->  (
s  =  ( F `
 g )  <->  S  =  ( F `  g ) ) )
54rexbidv 2868 . . . 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 3177 . . 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 474 . 2  |-  ( ( S  e.  C  /\  C  e.  (NoeACS `  X
) )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
87ancoms 453 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 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800    i^i cin 3438   ~Pcpw 3971   ` cfv 5529   Fincfn 7423  mrClscmrc 14643  ACScacs 14645  NoeACScnacs 29206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-nacs 29207
This theorem is referenced by:  isnacs3  29214
  Copyright terms: Public domain W3C validator