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Theorem isnacs2 26760
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 26758 . 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 13877 . . . . . . . . 9  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
41mrcf 13834 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
5 ffn 5591 . . . . . . . . 9  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
63, 4, 53syl 19 . . . . . . . 8  |-  ( C  e.  (ACS `  X
)  ->  F  Fn  ~P X )
7 inss1 3561 . . . . . . . 8  |-  ( ~P X  i^i  Fin )  C_ 
~P X
8 fvelimab 5782 . . . . . . . 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 644 . . . . . . 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 2438 . . . . . . . 8  |-  ( s  =  ( F `  g )  <->  ( F `  g )  =  s )
1110rexbii 2730 . . . . . . 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 256 . . . . . 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 2725 . . . . 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 3338 . . . . 5  |-  ( C 
C_  ( F "
( ~P X  i^i  Fin ) )  <->  A. s  e.  C  s  e.  ( F " ( ~P X  i^i  Fin )
) )
1513, 14syl6bbr 255 . . . 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 5216 . . . . . . 7  |-  ( F
" ( ~P X  i^i  Fin ) )  C_  ran  F
17 frn 5597 . . . . . . . 8  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
183, 4, 173syl 19 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ran  F  C_  C )
1916, 18syl5ss 3359 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  ( F " ( ~P X  i^i  Fin ) )  C_  C
)
2019biantrurd 495 . . . . 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 3363 . . . . 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 255 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  ( C  C_  ( F " ( ~P X  i^i  Fin )
)  <->  ( F "
( ~P X  i^i  Fin ) )  =  C ) )
2315, 22bitrd 245 . . 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 619 . 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 241 1  |-  ( C  e.  (NoeACS `  X
)  <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin )
)  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454   Fincfn 7109  Moorecmre 13807  mrClscmrc 13808  ACScacs 13810  NoeACScnacs 26756
This theorem is referenced by:  nacsacs  26763
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-mre 13811  df-mrc 13812  df-acs 13814  df-nacs 26757
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