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Theorem mrefg2 30241
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 14865 . . . . . . . 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 14862 . . . . . . . . . 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 3514 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  X )
72, 6impbida 830 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( g  C_  X  <->  g  C_  ( F `  g )
) )
8 vex 3116 . . . . . . . 8  |-  g  e. 
_V
98elpw 4016 . . . . . . 7  |-  ( g  e.  ~P X  <->  g  C_  X )
108elpw 4016 . . . . . . 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 3687 . . . . 5  |-  ( g  e.  ( ~P X  i^i  Fin )  <->  ( g  e.  ~P X  /\  g  e.  Fin ) )
14 elin 3687 . . . . 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 4013 . . . . . . 7  |-  ( S  =  ( F `  g )  ->  ~P S  =  ~P ( F `  g )
)
1716ineq1d 3699 . . . . . 6  |-  ( S  =  ( F `  g )  ->  ( ~P S  i^i  Fin )  =  ( ~P ( F `  g )  i^i  Fin ) )
1817eleq2d 2537 . . . . 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 2969 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 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   ` cfv 5586   Fincfn 7513  Moorecmre 14830  mrClscmrc 14831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-mre 14834  df-mrc 14835
This theorem is referenced by:  mrefg3  30242  isnacs3  30244
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