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Theorem isnacs2 26884
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 26882 . 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 13570 . . . . . . . . 9  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
41mrcf 13527 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
5 ffn 5405 . . . . . . . . 9  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
63, 4, 53syl 18 . . . . . . . 8  |-  ( C  e.  (ACS `  X
)  ->  F  Fn  ~P X )
7 inss1 3402 . . . . . . . 8  |-  ( ~P X  i^i  Fin )  C_ 
~P X
8 fvelimab 5594 . . . . . . . 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 643 . . . . . . 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 2298 . . . . . . . 8  |-  ( s  =  ( F `  g )  <->  ( F `  g )  =  s )
1110rexbii 2581 . . . . . . 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 255 . . . . . 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 2576 . . . . 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 3183 . . . . 5  |-  ( C 
C_  ( F "
( ~P X  i^i  Fin ) )  <->  A. s  e.  C  s  e.  ( F " ( ~P X  i^i  Fin )
) )
1513, 14syl6bbr 254 . . . 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 5041 . . . . . . 7  |-  ( F
" ( ~P X  i^i  Fin ) )  C_  ran  F
17 frn 5411 . . . . . . . 8  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
183, 4, 173syl 18 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ran  F  C_  C )
1916, 18syl5ss 3203 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  ( F " ( ~P X  i^i  Fin ) )  C_  C
)
2019biantrurd 494 . . . . 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 3207 . . . . 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 254 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  ( C  C_  ( F " ( ~P X  i^i  Fin )
)  <->  ( F "
( ~P X  i^i  Fin ) )  =  C ) )
2315, 22bitrd 244 . . 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 618 . 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 240 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   Fincfn 6879  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503  NoeACScnacs 26880
This theorem is referenced by:  nacsacs  26887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505  df-acs 13507  df-nacs 26881
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