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Theorem isnacs 26442
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 5691 . 2  |-  ( C  e.  (NoeACS `  X
)  ->  X  e.  _V )
2 elfvex 5691 . . 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 5661 . . . . . 6  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
5 pweq 3738 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65ineq1d 3477 . . . . . . . 8  |-  ( x  =  X  ->  ( ~P x  i^i  Fin )  =  ( ~P X  i^i  Fin ) )
76rexeqdv 2847 . . . . . . 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 2662 . . . . . 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 2887 . . . . 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 26441 . . . . 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 5675 . . . . . 6  |-  (ACS `  X )  e.  _V
1211rabex 4288 . . . . 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 5738 . . . 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 2447 . . 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 5661 . . . . . . . . 9  |-  ( c  =  C  ->  (mrCls `  c )  =  (mrCls `  C ) )
16 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
1715, 16syl6eqr 2430 . . . . . . . 8  |-  ( c  =  C  ->  (mrCls `  c )  =  F )
1817fveq1d 5663 . . . . . . 7  |-  ( c  =  C  ->  (
(mrCls `  c ) `  g )  =  ( F `  g ) )
1918eqeq2d 2391 . . . . . 6  |-  ( c  =  C  ->  (
s  =  ( (mrCls `  c ) `  g
)  <->  s  =  ( F `  g ) ) )
2019rexbidv 2663 . . . . 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 2848 . . . 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 3028 . . 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 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   {crab 2646   _Vcvv 2892    i^i cin 3255   ~Pcpw 3735   ` cfv 5387   Fincfn 7038  mrClscmrc 13728  ACScacs 13730  NoeACScnacs 26440
This theorem is referenced by:  nacsfg  26443  isnacs2  26444  isnacs3  26448  islnr3  26981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-nacs 26441
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