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Theorem submrc 13530
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 13507 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
213adant3 975 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( C  i^i  ~P D )  e.  (Moore `  D
) )
3 simp1 955 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  C  e.  (Moore `  X )
)
4 simp3 957 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  D )
5 mress 13495 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C )  ->  D  C_  X )
653adant3 975 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  D  C_  X )
74, 6sstrd 3189 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  X )
8 submrc.f . . . . 5  |-  F  =  (mrCls `  C )
98mrcssid 13519 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U
) )
103, 7, 9syl2anc 642 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( F `  U
) )
118mrccl 13513 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
123, 7, 11syl2anc 642 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  C )
138mrcsscl 13522 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  D  /\  D  e.  C )  ->  ( F `  U )  C_  D )
14133com23 1157 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  D )
15 fvex 5539 . . . . . 6  |-  ( F `
 U )  e. 
_V
1615elpw 3631 . . . . 5  |-  ( ( F `  U )  e.  ~P D  <->  ( F `  U )  C_  D
)
1714, 16sylibr 203 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ~P D )
18 elin 3358 . . . 4  |-  ( ( F `  U )  e.  ( C  i^i  ~P D )  <->  ( ( F `  U )  e.  C  /\  ( F `  U )  e.  ~P D ) )
1912, 17, 18sylanbrc 645 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  e.  ( C  i^i  ~P D ) )
20 submrc.g . . . 4  |-  G  =  (mrCls `  ( C  i^i  ~P D ) )
2120mrcsscl 13522 . . 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
) )
222, 10, 19, 21syl3anc 1182 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  C_  ( F `  U
) )
2320mrcssid 13519 . . . 4  |-  ( ( ( C  i^i  ~P D )  e.  (Moore `  D )  /\  U  C_  D )  ->  U  C_  ( G `  U
) )
242, 4, 23syl2anc 642 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  ( G `  U
) )
25 inss1 3389 . . . 4  |-  ( C  i^i  ~P D ) 
C_  C
2620mrccl 13513 . . . . 5  |-  ( ( ( C  i^i  ~P D )  e.  (Moore `  D )  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
272, 4, 26syl2anc 642 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  ( C  i^i  ~P D ) )
2825, 27sseldi 3178 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  C )
298mrcsscl 13522 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ( G `  U
)  /\  ( G `  U )  e.  C
)  ->  ( F `  U )  C_  ( G `  U )
)
303, 24, 28, 29syl3anc 1182 . 2  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( F `  U )  C_  ( G `  U
) )
3122, 30eqssd 3196 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 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  evlseu  19400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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