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Theorem isnacs 26779
Description: Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
isnacs  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
Distinct variable groups:    C, g,
s    g, F, s    g, X, s

Proof of Theorem isnacs
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5555 . 2  |-  ( C  e.  (NoeACS `  X
)  ->  X  e.  _V )
2 elfvex 5555 . . 3  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
32adantr 451 . 2  |-  ( ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )  ->  X  e.  _V )
4 fveq2 5525 . . . . . 6  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
5 pweq 3628 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65ineq1d 3369 . . . . . . . 8  |-  ( x  =  X  ->  ( ~P x  i^i  Fin )  =  ( ~P X  i^i  Fin ) )
76rexeqdv 2743 . . . . . . 7  |-  ( x  =  X  ->  ( E. g  e.  ( ~P x  i^i  Fin )
s  =  ( (mrCls `  c ) `  g
)  <->  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) ) )
87ralbidv 2563 . . . . . 6  |-  ( x  =  X  ->  ( A. s  e.  c  E. g  e.  ( ~P x  i^i  Fin )
s  =  ( (mrCls `  c ) `  g
)  <->  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) ) )
94, 8rabeqbidv 2783 . . . . 5  |-  ( x  =  X  ->  { c  e.  (ACS `  x
)  |  A. s  e.  c  E. g  e.  ( ~P x  i^i 
Fin ) s  =  ( (mrCls `  c
) `  g ) }  =  { c  e.  (ACS `  X )  |  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) } )
10 df-nacs 26778 . . . . 5  |- NoeACS  =  ( x  e.  _V  |->  { c  e.  (ACS `  x )  |  A. s  e.  c  E. g  e.  ( ~P x  i^i  Fin ) s  =  ( (mrCls `  c ) `  g
) } )
11 fvex 5539 . . . . . 6  |-  (ACS `  X )  e.  _V
1211rabex 4165 . . . . 5  |-  { c  e.  (ACS `  X
)  |  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) }  e.  _V
139, 10, 12fvmpt 5602 . . . 4  |-  ( X  e.  _V  ->  (NoeACS `  X )  =  {
c  e.  (ACS `  X )  |  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g
) } )
1413eleq2d 2350 . . 3  |-  ( X  e.  _V  ->  ( C  e.  (NoeACS `  X
)  <->  C  e.  { c  e.  (ACS `  X
)  |  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) } ) )
15 fveq2 5525 . . . . . . . . 9  |-  ( c  =  C  ->  (mrCls `  c )  =  (mrCls `  C ) )
16 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
1715, 16syl6eqr 2333 . . . . . . . 8  |-  ( c  =  C  ->  (mrCls `  c )  =  F )
1817fveq1d 5527 . . . . . . 7  |-  ( c  =  C  ->  (
(mrCls `  c ) `  g )  =  ( F `  g ) )
1918eqeq2d 2294 . . . . . 6  |-  ( c  =  C  ->  (
s  =  ( (mrCls `  c ) `  g
)  <->  s  =  ( F `  g ) ) )
2019rexbidv 2564 . . . . 5  |-  ( c  =  C  ->  ( E. g  e.  ( ~P X  i^i  Fin )
s  =  ( (mrCls `  c ) `  g
)  <->  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) ) )
2120raleqbi1dv 2744 . . . 4  |-  ( c  =  C  ->  ( A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( (mrCls `  c ) `  g
)  <->  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
2221elrab 2923 . . 3  |-  ( C  e.  { c  e.  (ACS `  X )  |  A. s  e.  c  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) }  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
2314, 22syl6bb 252 . 2  |-  ( X  e.  _V  ->  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) ) )
241, 3, 23pm5.21nii 342 1  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625   ` cfv 5255   Fincfn 6863  mrClscmrc 13485  ACScacs 13487  NoeACScnacs 26777
This theorem is referenced by:  nacsfg  26780  isnacs2  26781  isnacs3  26785  islnr3  27319
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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