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Theorem isnacs2 30800
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 30798 . 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 15068 . . . . . . . . 9  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
41mrcf 15025 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
5 ffn 5737 . . . . . . . . 9  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
63, 4, 53syl 20 . . . . . . . 8  |-  ( C  e.  (ACS `  X
)  ->  F  Fn  ~P X )
7 inss1 3714 . . . . . . . 8  |-  ( ~P X  i^i  Fin )  C_ 
~P X
8 fvelimab 5929 . . . . . . . 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 2466 . . . . . . . 8  |-  ( s  =  ( F `  g )  <->  ( F `  g )  =  s )
1110rexbii 2959 . . . . . . 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 2896 . . . . 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 3489 . . . . 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 5358 . . . . . . 7  |-  ( F
" ( ~P X  i^i  Fin ) )  C_  ran  F
17 frn 5743 . . . . . . . 8  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
183, 4, 173syl 20 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ran  F  C_  C )
1916, 18syl5ss 3510 . . . . . 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 3514 . . . . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594   Fincfn 7535  Moorecmre 14998  mrClscmrc 14999  ACScacs 15001  NoeACScnacs 30796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 15002  df-mrc 15003  df-acs 15005  df-nacs 30797
This theorem is referenced by:  nacsacs  30803
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