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Theorem submrc 13854
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 13831 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
213adant3 978 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
3 simp1 958 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  C  e.  (Moore `  X )
)
4 submrc.f . . . 4  |-  F  =  (mrCls `  C )
5 simp3 960 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  D )
6 mress 13819 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  D  C_  X )
763adant3 978 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  D  C_  X )
85, 7sstrd 3359 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  X )
93, 4, 8mrcssidd 13851 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( F `  U
) )
104mrccl 13837 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
113, 8, 10syl2anc 644 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  C )
124mrcsscl 13846 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  D  /\  D  e.  C )  ->  ( F `  U )  C_  D )
13123com23 1160 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  D )
14 fvex 5743 . . . . . 6  |-  ( F `
 U )  e. 
_V
1514elpw 3806 . . . . 5  |-  ( ( F `  U )  e.  ~P D  <->  ( F `  U )  C_  D
)
1613, 15sylibr 205 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ~P D )
17 elin 3531 . . . 4  |-  ( ( F `  U )  e.  ( C  i^i  ~P D )  <->  ( ( F `  U )  e.  C  /\  ( F `  U )  e.  ~P D ) )
1811, 16, 17sylanbrc 647 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ( C  i^i  ~P D ) )
19 submrc.g . . . 4  |-  G  =  (mrCls `  ( C  i^i  ~P D ) )
2019mrcsscl 13846 . . 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
) )
212, 9, 18, 20syl3anc 1185 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  C_  ( F `  U
) )
222, 19, 5mrcssidd 13851 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( G `  U
) )
23 inss1 3562 . . . 4  |-  ( C  i^i  ~P D ) 
C_  C
2419mrccl 13837 . . . . 5  |-  ( ( ( C  i^i  ~P D )  e.  (Moore `  D )  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
252, 5, 24syl2anc 644 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
2623, 25sseldi 3347 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  C )
274mrcsscl 13846 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ( G `  U
)  /\  ( G `  U )  e.  C
)  ->  ( F `  U )  C_  ( G `  U )
)
283, 22, 26, 27syl3anc 1185 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  ( G `  U
) )
2921, 28eqssd 3366 1  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320    C_ wss 3321   ~Pcpw 3800   ` cfv 5455  Moorecmre 13808  mrClscmrc 13809
This theorem is referenced by:  evlseu  19938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-mre 13812  df-mrc 13813
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