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Theorem mrefg2 26885
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 13535 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  X )  ->  g  C_  ( F `  g
) )
3 simpr 447 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  ( F `  g
) )
41mrcssv 13532 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F `  g )  C_  X
)
54adantr 451 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  ( F `  g )  C_  X )
63, 5sstrd 3202 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  X )
72, 6impbida 805 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( g  C_  X  <->  g  C_  ( F `  g )
) )
8 vex 2804 . . . . . . . 8  |-  g  e. 
_V
98elpw 3644 . . . . . . 7  |-  ( g  e.  ~P X  <->  g  C_  X )
108elpw 3644 . . . . . . 7  |-  ( g  e.  ~P ( F `
 g )  <->  g  C_  ( F `  g ) )
117, 9, 103bitr4g 279 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ~P X  <->  g  e.  ~P ( F `  g
) ) )
1211anbi1d 685 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( (
g  e.  ~P X  /\  g  e.  Fin ) 
<->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) ) )
13 elin 3371 . . . . 5  |-  ( g  e.  ( ~P X  i^i  Fin )  <->  ( g  e.  ~P X  /\  g  e.  Fin ) )
14 elin 3371 . . . . 5  |-  ( g  e.  ( ~P ( F `  g )  i^i  Fin )  <->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) )
1512, 13, 143bitr4g 279 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
16 pweq 3641 . . . . . . 7  |-  ( S  =  ( F `  g )  ->  ~P S  =  ~P ( F `  g )
)
1716ineq1d 3382 . . . . . 6  |-  ( S  =  ( F `  g )  ->  ( ~P S  i^i  Fin )  =  ( ~P ( F `  g )  i^i  Fin ) )
1817eleq2d 2363 . . . . 5  |-  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P S  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
1918bibi2d 309 . . . 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 213 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P S  i^i  Fin ) ) ) )
2120pm5.32rd 621 . 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 2579 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   ` cfv 5271   Fincfn 6879  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrefg3  26886  isnacs3  26888
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-csb 3095  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
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