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Theorem submrc 14587
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 14564 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
213adant3 1008 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
3 simp1 988 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  C  e.  (Moore `  X )
)
4 submrc.f . . . 4  |-  F  =  (mrCls `  C )
5 simp3 990 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  D )
6 mress 14552 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  D  C_  X )
763adant3 1008 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  D  C_  X )
85, 7sstrd 3387 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  X )
93, 4, 8mrcssidd 14584 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( F `  U
) )
104mrccl 14570 . . . . 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 14579 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  D  /\  D  e.  C )  ->  ( F `  U )  C_  D )
13123com23 1193 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  D )
14 fvex 5722 . . . . . 6  |-  ( F `
 U )  e. 
_V
1514elpw 3887 . . . . 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 3561 . . 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 14579 . . 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 1218 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  C_  ( F `  U
) )
212, 18, 5mrcssidd 14584 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( G `  U
) )
22 inss1 3591 . . . 4  |-  ( C  i^i  ~P D ) 
C_  C
2318mrccl 14570 . . . . 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 3375 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  C )
264mrcsscl 14579 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ( G `  U
)  /\  ( G `  U )  e.  C
)  ->  ( F `  U )  C_  ( G `  U )
)
273, 21, 25, 26syl3anc 1218 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  ( G `  U
) )
2820, 27eqssd 3394 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 965    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   ` cfv 5439  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:  evlseu  17624
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