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Theorem submrc 15044
Description: In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
submrc.f  |-  F  =  (mrCls `  C )
submrc.g  |-  G  =  (mrCls `  ( C  i^i  ~P D ) )
Assertion
Ref Expression
submrc  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )

Proof of Theorem submrc
StepHypRef Expression
1 submre 15021 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
213adant3 1016 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
3 simp1 996 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  C  e.  (Moore `  X )
)
4 submrc.f . . . 4  |-  F  =  (mrCls `  C )
5 simp3 998 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  D )
6 mress 15009 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  D  C_  X )
763adant3 1016 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  D  C_  X )
85, 7sstrd 3509 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  X )
93, 4, 8mrcssidd 15041 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( F `  U
) )
104mrccl 15027 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
113, 8, 10syl2anc 661 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  C )
124mrcsscl 15036 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  D  /\  D  e.  C )  ->  ( F `  U )  C_  D )
13123com23 1202 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  D )
14 fvex 5882 . . . . . 6  |-  ( F `
 U )  e. 
_V
1514elpw 4021 . . . . 5  |-  ( ( F `  U )  e.  ~P D  <->  ( F `  U )  C_  D
)
1613, 15sylibr 212 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ~P D )
1711, 16elind 3684 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ( C  i^i  ~P D ) )
18 submrc.g . . . 4  |-  G  =  (mrCls `  ( C  i^i  ~P D ) )
1918mrcsscl 15036 . . 3  |-  ( ( ( C  i^i  ~P D )  e.  (Moore `  D )  /\  U  C_  ( F `  U
)  /\  ( F `  U )  e.  ( C  i^i  ~P D
) )  ->  ( G `  U )  C_  ( F `  U
) )
202, 9, 17, 19syl3anc 1228 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  C_  ( F `  U
) )
212, 18, 5mrcssidd 15041 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( G `  U
) )
22 inss1 3714 . . . 4  |-  ( C  i^i  ~P D ) 
C_  C
2318mrccl 15027 . . . . 5  |-  ( ( ( C  i^i  ~P D )  e.  (Moore `  D )  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
242, 5, 23syl2anc 661 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
2522, 24sseldi 3497 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  C )
264mrcsscl 15036 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ( G `  U
)  /\  ( G `  U )  e.  C
)  ->  ( F `  U )  C_  ( G `  U )
)
273, 21, 25, 26syl3anc 1228 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  ( G `  U
) )
2820, 27eqssd 3516 1  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   ` cfv 5594  Moorecmre 14998  mrClscmrc 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 15002  df-mrc 15003
This theorem is referenced by:  evlseu  18311
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