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Theorem isnacs 30268
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 5893 . 2  |-  ( C  e.  (NoeACS `  X
)  ->  X  e.  _V )
2 elfvex 5893 . . 3  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
32adantr 465 . 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 5866 . . . . . 6  |-  ( x  =  X  ->  (ACS `  x )  =  (ACS
`  X ) )
5 pweq 4013 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65ineq1d 3699 . . . . . . . 8  |-  ( x  =  X  ->  ( ~P x  i^i  Fin )  =  ( ~P X  i^i  Fin ) )
76rexeqdv 3065 . . . . . . 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 2903 . . . . . 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 3108 . . . . 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 30267 . . . . 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 5876 . . . . . 6  |-  (ACS `  X )  e.  _V
1211rabex 4598 . . . . 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 5950 . . . 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 2537 . . 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 5866 . . . . . . . . 9  |-  ( c  =  C  ->  (mrCls `  c )  =  (mrCls `  C ) )
16 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
1715, 16syl6eqr 2526 . . . . . . . 8  |-  ( c  =  C  ->  (mrCls `  c )  =  F )
1817fveq1d 5868 . . . . . . 7  |-  ( c  =  C  ->  (
(mrCls `  c ) `  g )  =  ( F `  g ) )
1918eqeq2d 2481 . . . . . 6  |-  ( c  =  C  ->  (
s  =  ( (mrCls `  c ) `  g
)  <->  s  =  ( F `  g ) ) )
2019rexbidv 2973 . . . . 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 3066 . . . 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 3261 . . 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 261 . 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 353 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    i^i cin 3475   ~Pcpw 4010   ` cfv 5588   Fincfn 7516  mrClscmrc 14838  ACScacs 14840  NoeACScnacs 30266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-nacs 30267
This theorem is referenced by:  nacsfg  30269  isnacs2  30270  isnacs3  30274  islnr3  30696
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