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Theorem mrefg3 35015
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 35014 . . 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 754 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  C  e.  (Moore `  X )
)
5 inss1 3661 . . . . . . . . 9  |-  ( ~P S  i^i  Fin )  C_ 
~P S
65sseli 3440 . . . . . . . 8  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  e.  ~P S )
76elpwid 3967 . . . . . . 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 756 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  S  e.  C )
101mrcsscl 15236 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  S  /\  S  e.  C )  ->  ( F `  g )  C_  S )
114, 8, 9, 10syl3anc 1232 . . . . 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 3459 . . . 4  |-  ( S  =  ( F `  g )  <->  ( S  C_  ( F `  g
)  /\  ( F `  g )  C_  S
) )
1412, 13syl6rbbr 266 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  S  e.  C )  /\  g  e.  ( ~P S  i^i  Fin ) )  ->  ( S  =  ( F `  g )  <->  S  C_  ( F `  g )
) )
1514rexbidva 2917 . 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 255 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 186    /\ wa 369    = wceq 1407    e. wcel 1844   E.wrex 2757    i^i cin 3415    C_ wss 3416   ~Pcpw 3957   ` cfv 5571   Fincfn 7556  Moorecmre 15198  mrClscmrc 15199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-mre 15202  df-mrc 15203
This theorem is referenced by: (None)
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