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Theorem isnacs 35284
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 5899 . 2  |-  ( C  e.  (NoeACS `  X
)  ->  X  e.  _V )
2 elfvex 5899 . . 3  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
32adantr 466 . 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 5872 . . . . . 6  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
5 pweq 3979 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65ineq1d 3660 . . . . . . . 8  |-  ( x  =  X  ->  ( ~P x  i^i  Fin )  =  ( ~P X  i^i  Fin ) )
76rexeqdv 3030 . . . . . . 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 2862 . . . . . 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 3073 . . . . 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 35283 . . . . 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 5882 . . . . . 6  |-  (ACS `  X )  e.  _V
1211rabex 4567 . . . . 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 5955 . . . 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 2490 . . 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 5872 . . . . . . . . 9  |-  ( c  =  C  ->  (mrCls `  c )  =  (mrCls `  C ) )
16 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
1715, 16syl6eqr 2479 . . . . . . . 8  |-  ( c  =  C  ->  (mrCls `  c )  =  F )
1817fveq1d 5874 . . . . . . 7  |-  ( c  =  C  ->  (
(mrCls `  c ) `  g )  =  ( F `  g ) )
1918eqeq2d 2434 . . . . . 6  |-  ( c  =  C  ->  (
s  =  ( (mrCls `  c ) `  g
)  <->  s  =  ( F `  g ) ) )
2019rexbidv 2937 . . . . 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 3031 . . . 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 3226 . . 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 264 . 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 354 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 setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078    i^i cin 3432   ~Pcpw 3976   ` cfv 5592   Fincfn 7568  mrClscmrc 15433  ACScacs 15435  NoeACScnacs 35282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-nacs 35283
This theorem is referenced by:  nacsfg  35285  isnacs2  35286  isnacs3  35290  islnr3  35713
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