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Theorem submrc 13546
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 13523 . . . 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 13511 . . . . . 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 3202 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  U  C_  X )
8 submrc.f . . . . 5  |-  F  =  (mrCls `  C )
98mrcssid 13535 . . . 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 13529 . . . . 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 13538 . . . . . 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 5555 . . . . . 6  |-  ( F `
 U )  e. 
_V
1615elpw 3644 . . . . 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 3371 . . . 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 13538 . . 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 13535 . . . 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 3402 . . . 4  |-  ( C  i^i  ~P D ) 
C_  C
2620mrccl 13529 . . . . 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 3191 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  e.  C )
298mrcsscl 13538 . . 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 3209 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  evlseu  19416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
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