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Theorem isnacs 26749
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 5750 . 2  |-  ( C  e.  (NoeACS `  X
)  ->  X  e.  _V )
2 elfvex 5750 . . 3  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
32adantr 452 . 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 5720 . . . . . 6  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
5 pweq 3794 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65ineq1d 3533 . . . . . . . 8  |-  ( x  =  X  ->  ( ~P x  i^i  Fin )  =  ( ~P X  i^i  Fin ) )
76rexeqdv 2903 . . . . . . 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 2717 . . . . . 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 2943 . . . . 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 26748 . . . . 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 5734 . . . . . 6  |-  (ACS `  X )  e.  _V
1211rabex 4346 . . . . 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 5798 . . . 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 2502 . . 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 5720 . . . . . . . . 9  |-  ( c  =  C  ->  (mrCls `  c )  =  (mrCls `  C ) )
16 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
1715, 16syl6eqr 2485 . . . . . . . 8  |-  ( c  =  C  ->  (mrCls `  c )  =  F )
1817fveq1d 5722 . . . . . . 7  |-  ( c  =  C  ->  (
(mrCls `  c ) `  g )  =  ( F `  g ) )
1918eqeq2d 2446 . . . . . 6  |-  ( c  =  C  ->  (
s  =  ( (mrCls `  c ) `  g
)  <->  s  =  ( F `  g ) ) )
2019rexbidv 2718 . . . . 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 2904 . . . 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 3084 . . 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 253 . 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 343 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    i^i cin 3311   ~Pcpw 3791   ` cfv 5446   Fincfn 7101  mrClscmrc 13800  ACScacs 13802  NoeACScnacs 26747
This theorem is referenced by:  nacsfg  26750  isnacs2  26751  isnacs3  26755  islnr3  27287
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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