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Theorem mrefg3 26783
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 26782 . . 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 451 . 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 730 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  C  e.  (Moore `  X )
)
5 inss1 3389 . . . . . . . . 9  |-  ( ~P S  i^i  Fin )  C_ 
~P S
65sseli 3176 . . . . . . . 8  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  e.  ~P S )
7 elpwi 3633 . . . . . . . 8  |-  ( g  e.  ~P S  -> 
g  C_  S )
86, 7syl 15 . . . . . . 7  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  C_  S )
98adantl 452 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  g  C_  S )
10 simplr 731 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  S  e.  C )
111mrcsscl 13522 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  S  /\  S  e.  C )  ->  ( F `  g )  C_  S )
124, 9, 10, 11syl3anc 1182 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( F `  g )  C_  S )
1312biantrud 493 . . . 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
) ) )
14 eqss 3194 . . . 4  |-  ( S  =  ( F `  g )  <->  ( S  C_  ( F `  g
)  /\  ( F `  g )  C_  S
) )
1513, 14syl6rbbr 255 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( S  =  ( F `  g )  <->  S  C_  ( F `  g )
) )
1615rexbidva 2560 . 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 )
) )
173, 16bitrd 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Fincfn 6863  Moorecmre 13484  mrClscmrc 13485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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