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Theorem isnacs2 26781
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 26779 . 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 13554 . . . . . . . . 9  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
41mrcf 13511 . . . . . . . . 9  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
5 ffn 5389 . . . . . . . . 9  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
63, 4, 53syl 18 . . . . . . . 8  |-  ( C  e.  (ACS `  X
)  ->  F  Fn  ~P X )
7 inss1 3389 . . . . . . . 8  |-  ( ~P X  i^i  Fin )  C_ 
~P X
8 fvelimab 5578 . . . . . . . 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 2285 . . . . . . . 8  |-  ( s  =  ( F `  g )  <->  ( F `  g )  =  s )
1110rexbii 2568 . . . . . . 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 2563 . . . . 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 3170 . . . . 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 5025 . . . . . . 7  |-  ( F
" ( ~P X  i^i  Fin ) )  C_  ran  F
17 frn 5395 . . . . . . . 8  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
183, 4, 173syl 18 . . . . . . 7  |-  ( C  e.  (ACS `  X
)  ->  ran  F  C_  C )
1916, 18syl5ss 3190 . . . . . 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 3194 . . . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   Fincfn 6863  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487  NoeACScnacs 26777
This theorem is referenced by:  nacsacs  26784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489  df-acs 13491  df-nacs 26778
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