Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrefg3 Structured version   Unicode version

Theorem mrefg3 29070
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 29069 . . 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 3591 . . . . . . . . 9  |-  ( ~P S  i^i  Fin )  C_ 
~P S
65sseli 3373 . . . . . . . 8  |-  ( g  e.  ( ~P S  i^i  Fin )  ->  g  e.  ~P S )
76elpwid 3891 . . . . . . 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 14579 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  g  C_  S  /\  S  e.  C )  ->  ( F `  g )  C_  S )
114, 8, 9, 10syl3anc 1218 . . . . 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 3392 . . . 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 2753 . 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 1369    e. wcel 1756   E.wrex 2737    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   ` cfv 5439   Fincfn 7331  Moorecmre 14541  mrClscmrc 14542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-mre 14545  df-mrc 14546
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator