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Theorem mrefg3 26762
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
mrefg3  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  C_  ( F `  g )
) )
Distinct variable groups:    C, g    g, F    S, g    g, X

Proof of Theorem mrefg3
StepHypRef Expression
1 isnacs.f . . . 4  |-  F  =  (mrCls `  C )
21mrefg2 26761 . . 3  |-  ( 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 ) ) )
32adantr 452 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g ) ) )
4 simpll 731 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  C  e.  (Moore `  X )
)
5 inss1 3561 . . . . . . . . 9  |-  ( ~P S  i^i  Fin )  C_ 
~P S
65sseli 3344 . . . . . . . 8  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  e.  ~P S )
76elpwid 3808 . . . . . . 7  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  C_  S )
87adantl 453 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  g  C_  S )
9 simplr 732 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  S  e.  C )
101mrcsscl 13845 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  S  /\  S  e.  C )  ->  ( F `  g )  C_  S )
114, 8, 9, 10syl3anc 1184 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( F `  g )  C_  S )
1211biantrud 494 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( S  C_  ( F `  g )  <->  ( S  C_  ( F `  g
)  /\  ( F `  g )  C_  S
) ) )
13 eqss 3363 . . . 4  |-  ( S  =  ( F `  g )  <->  ( S  C_  ( F `  g
)  /\  ( F `  g )  C_  S
) )
1412, 13syl6rbbr 256 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( S  =  ( F `  g )  <->  S  C_  ( F `  g )
) )
1514rexbidva 2722 . 2  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  C_  ( F `  g )
) )
163, 15bitrd 245 1  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  C_  ( F `  g )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   ` cfv 5454   Fincfn 7109  Moorecmre 13807  mrClscmrc 13808
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-mre 13811  df-mrc 13812
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