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Theorem mrefg2 29043
Description: Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrefg2  |-  ( C  e.  (Moore `  X
)  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g ) ) )
Distinct variable groups:    C, g    g, F    S, g    g, X

Proof of Theorem mrefg2
StepHypRef Expression
1 isnacs.f . . . . . . . . 9  |-  F  =  (mrCls `  C )
21mrcssid 14555 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  X )  ->  g  C_  ( F `  g
) )
3 simpr 461 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  ( F `  g
) )
41mrcssv 14552 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F `  g )  C_  X
)
54adantr 465 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  ( F `  g )  C_  X )
63, 5sstrd 3366 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  X )
72, 6impbida 828 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( g  C_  X  <->  g  C_  ( F `  g )
) )
8 vex 2975 . . . . . . . 8  |-  g  e. 
_V
98elpw 3866 . . . . . . 7  |-  ( g  e.  ~P X  <->  g  C_  X )
108elpw 3866 . . . . . . 7  |-  ( g  e.  ~P ( F `
 g )  <->  g  C_  ( F `  g ) )
117, 9, 103bitr4g 288 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ~P X  <->  g  e.  ~P ( F `  g
) ) )
1211anbi1d 704 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( (
g  e.  ~P X  /\  g  e.  Fin ) 
<->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) ) )
13 elin 3539 . . . . 5  |-  ( g  e.  ( ~P X  i^i  Fin )  <->  ( g  e.  ~P X  /\  g  e.  Fin ) )
14 elin 3539 . . . . 5  |-  ( g  e.  ( ~P ( F `  g )  i^i  Fin )  <->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) )
1512, 13, 143bitr4g 288 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
16 pweq 3863 . . . . . . 7  |-  ( S  =  ( F `  g )  ->  ~P S  =  ~P ( F `  g )
)
1716ineq1d 3551 . . . . . 6  |-  ( S  =  ( F `  g )  ->  ( ~P S  i^i  Fin )  =  ( ~P ( F `  g )  i^i  Fin ) )
1817eleq2d 2510 . . . . 5  |-  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P S  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
1918bibi2d 318 . . . 4  |-  ( S  =  ( F `  g )  ->  (
( g  e.  ( ~P X  i^i  Fin ) 
<->  g  e.  ( ~P S  i^i  Fin )
)  <->  ( g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) ) )
2015, 19syl5ibrcom 222 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P S  i^i  Fin ) ) ) )
2120pm5.32rd 640 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( (
g  e.  ( ~P X  i^i  Fin )  /\  S  =  ( F `  g )
)  <->  ( g  e.  ( ~P S  i^i  Fin )  /\  S  =  ( F `  g
) ) ) )
2221rexbidv2 2738 1  |-  ( C  e.  (Moore `  X
)  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   ` cfv 5418   Fincfn 7310  Moorecmre 14520  mrClscmrc 14521
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-mre 14524  df-mrc 14525
This theorem is referenced by:  mrefg3  29044  isnacs3  29046
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