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Theorem mrefg2 26752
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 13834 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  X )  ->  g  C_  ( F `  g
) )
3 simpr 448 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  ( F `  g
) )
41mrcssv 13831 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F `  g )  C_  X
)
54adantr 452 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  ( F `  g )  C_  X )
63, 5sstrd 3350 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  ( F `  g
) )  ->  g  C_  X )
72, 6impbida 806 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  ( g  C_  X  <->  g  C_  ( F `  g )
) )
8 vex 2951 . . . . . . . 8  |-  g  e. 
_V
98elpw 3797 . . . . . . 7  |-  ( g  e.  ~P X  <->  g  C_  X )
108elpw 3797 . . . . . . 7  |-  ( g  e.  ~P ( F `
 g )  <->  g  C_  ( F `  g ) )
117, 9, 103bitr4g 280 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ~P X  <->  g  e.  ~P ( F `  g
) ) )
1211anbi1d 686 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( (
g  e.  ~P X  /\  g  e.  Fin ) 
<->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) ) )
13 elin 3522 . . . . 5  |-  ( g  e.  ( ~P X  i^i  Fin )  <->  ( g  e.  ~P X  /\  g  e.  Fin ) )
14 elin 3522 . . . . 5  |-  ( g  e.  ( ~P ( F `  g )  i^i  Fin )  <->  ( g  e.  ~P ( F `  g )  /\  g  e.  Fin ) )
1512, 13, 143bitr4g 280 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
16 pweq 3794 . . . . . . 7  |-  ( S  =  ( F `  g )  ->  ~P S  =  ~P ( F `  g )
)
1716ineq1d 3533 . . . . . 6  |-  ( S  =  ( F `  g )  ->  ( ~P S  i^i  Fin )  =  ( ~P ( F `  g )  i^i  Fin ) )
1817eleq2d 2502 . . . . 5  |-  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P S  i^i  Fin )  <->  g  e.  ( ~P ( F `  g )  i^i  Fin ) ) )
1918bibi2d 310 . . . 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 214 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( S  =  ( F `  g )  ->  (
g  e.  ( ~P X  i^i  Fin )  <->  g  e.  ( ~P S  i^i  Fin ) ) ) )
2120pm5.32rd 622 . 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 2720 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   ` cfv 5446   Fincfn 7101  Moorecmre 13799  mrClscmrc 13800
This theorem is referenced by:  mrefg3  26753  isnacs3  26755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-mre 13803  df-mrc 13804
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