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Theorem mrefg2 35001
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 15231 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  X )  ->  g  C_  ( F `  g
) )
3 simpr 459 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  ( F `  g
) )
41mrcssv 15228 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F `  g )  C_  X
)
54adantr 463 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  ( F `  g )  C_  X )
63, 5sstrd 3452 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  X )
72, 6impbida 833 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( g  C_  X  <->  g  C_  ( F `  g )
) )
8 vex 3062 . . . . . . . 8  |-  g  e. 
_V
98elpw 3961 . . . . . . 7  |-  ( g  e.  ~P X  <->  g  C_  X )
108elpw 3961 . . . . . . 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 703 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( (
g  e.  ~P X  /\  g  e.  Fin ) 
<->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) ) )
13 elin 3626 . . . . 5  |-  ( g  e.  ( ~P X  i^i  Fin )  <->  ( g  e.  ~P X  /\  g  e.  Fin ) )
14 elin 3626 . . . . 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 3958 . . . . . . 7  |-  ( S  =  ( F `  g )  ->  ~P S  =  ~P ( F `  g )
)
1716ineq1d 3640 . . . . . 6  |-  ( S  =  ( F `  g )  ->  ( ~P S  i^i  Fin )  =  ( ~P ( F `  g )  i^i  Fin ) )
1817eleq2d 2472 . . . . 5  |-  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P S  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
1918bibi2d 316 . . . 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 638 . 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 2914 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 367    = wceq 1405    e. wcel 1842   E.wrex 2755    i^i cin 3413    C_ wss 3414   ~Pcpw 3955   ` cfv 5569   Fincfn 7554  Moorecmre 15196  mrClscmrc 15197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-mre 15200  df-mrc 15201
This theorem is referenced by:  mrefg3  35002  isnacs3  35004
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