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Theorem mrefg3 30272
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 30271 . . 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 465 . 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 753 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  C  e.  (Moore `  X )
)
5 inss1 3718 . . . . . . . . 9  |-  ( ~P S  i^i  Fin )  C_ 
~P S
65sseli 3500 . . . . . . . 8  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  e.  ~P S )
76elpwid 4020 . . . . . . 7  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  C_  S )
87adantl 466 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  g  C_  S )
9 simplr 754 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  S  e.  C )
101mrcsscl 14875 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  S  /\  S  e.  C )  ->  ( F `  g )  C_  S )
114, 8, 9, 10syl3anc 1228 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( F `  g )  C_  S )
1211biantrud 507 . . . 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 3519 . . . 4  |-  ( S  =  ( F `  g )  <->  ( S  C_  ( F `  g
)  /\  ( F `  g )  C_  S
) )
1412, 13syl6rbbr 264 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( S  =  ( F `  g )  <->  S  C_  ( F `  g )
) )
1514rexbidva 2970 . 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 253 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 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 5588   Fincfn 7516  Moorecmre 14837  mrClscmrc 14838
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-mre 14841  df-mrc 14842
This theorem is referenced by: (None)
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