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Theorem isnacs2 29068
Description: Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
isnacs2  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin )
)  =  C ) )

Proof of Theorem isnacs2
Dummy variables  g 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . 3  |-  F  =  (mrCls `  C )
21isnacs 29066 . 2  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g ) ) )
3 acsmre 14611 . . . . . . . . 9  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
41mrcf 14568 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
5 ffn 5580 . . . . . . . . 9  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
63, 4, 53syl 20 . . . . . . . 8  |-  ( C  e.  (ACS `  X
)  ->  F  Fn  ~P X )
7 inss1 3591 . . . . . . . 8  |-  ( ~P X  i^i  Fin )  C_ 
~P X
8 fvelimab 5768 . . . . . . . 8  |-  ( ( F  Fn  ~P X  /\  ( ~P X  i^i  Fin )  C_  ~P X
)  ->  ( s  e.  ( F " ( ~P X  i^i  Fin )
)  <->  E. g  e.  ( ~P X  i^i  Fin ) ( F `  g )  =  s ) )
96, 7, 8sylancl 662 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ( s  e.  ( F " ( ~P X  i^i  Fin )
)  <->  E. g  e.  ( ~P X  i^i  Fin ) ( F `  g )  =  s ) )
10 eqcom 2445 . . . . . . . 8  |-  ( s  =  ( F `  g )  <->  ( F `  g )  =  s )
1110rexbii 2761 . . . . . . 7  |-  ( E. g  e.  ( ~P X  i^i  Fin )
s  =  ( F `
 g )  <->  E. g  e.  ( ~P X  i^i  Fin ) ( F `  g )  =  s )
129, 11syl6rbbr 264 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  ( E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g )  <->  s  e.  ( F " ( ~P X  i^i  Fin )
) ) )
1312ralbidv 2756 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g )  <->  A. s  e.  C  s  e.  ( F " ( ~P X  i^i  Fin )
) ) )
14 dfss3 3367 . . . . 5  |-  ( C 
C_  ( F "
( ~P X  i^i  Fin ) )  <->  A. s  e.  C  s  e.  ( F " ( ~P X  i^i  Fin )
) )
1513, 14syl6bbr 263 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  ( A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g )  <->  C  C_  ( F " ( ~P X  i^i  Fin ) ) ) )
16 imassrn 5201 . . . . . . 7  |-  ( F
" ( ~P X  i^i  Fin ) )  C_  ran  F
17 frn 5586 . . . . . . . 8  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
183, 4, 173syl 20 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ran  F  C_  C )
1916, 18syl5ss 3388 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  ( F " ( ~P X  i^i  Fin ) )  C_  C
)
2019biantrurd 508 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( C  C_  ( F " ( ~P X  i^i  Fin )
)  <->  ( ( F
" ( ~P X  i^i  Fin ) )  C_  C  /\  C  C_  ( F " ( ~P X  i^i  Fin ) ) ) ) )
21 eqss 3392 . . . . 5  |-  ( ( F " ( ~P X  i^i  Fin )
)  =  C  <->  ( ( F " ( ~P X  i^i  Fin ) )  C_  C  /\  C  C_  ( F " ( ~P X  i^i  Fin ) ) ) )
2220, 21syl6bbr 263 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  ( C  C_  ( F " ( ~P X  i^i  Fin )
)  <->  ( F "
( ~P X  i^i  Fin ) )  =  C ) )
2315, 22bitrd 253 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g )  <->  ( F " ( ~P X  i^i  Fin ) )  =  C ) )
2423pm5.32i 637 . 2  |-  ( ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) )  <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin )
)  =  C ) )
252, 24bitri 249 1  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin )
)  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   ran crn 4862   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439   Fincfn 7331  Moorecmre 14541  mrClscmrc 14542  ACScacs 14544  NoeACScnacs 29064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-mre 14545  df-mrc 14546  df-acs 14548  df-nacs 29065
This theorem is referenced by:  nacsacs  29071
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