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Theorem mrcss 13518
Description: Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcss  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )

Proof of Theorem mrcss
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 sstr2 3186 . . . . . 6  |-  ( U 
C_  V  ->  ( V  C_  s  ->  U  C_  s ) )
21adantr 451 . . . . 5  |-  ( ( U  C_  V  /\  s  e.  C )  ->  ( V  C_  s  ->  U  C_  s )
)
32ss2rabdv 3254 . . . 4  |-  ( U 
C_  V  ->  { s  e.  C  |  V  C_  s }  C_  { s  e.  C  |  U  C_  s } )
4 intss 3883 . . . 4  |-  ( { s  e.  C  |  V  C_  s }  C_  { s  e.  C  |  U  C_  s }  ->  |^|
{ s  e.  C  |  U  C_  s } 
C_  |^| { s  e.  C  |  V  C_  s } )
53, 4syl 15 . . 3  |-  ( U 
C_  V  ->  |^| { s  e.  C  |  U  C_  s }  C_  |^| { s  e.  C  |  V  C_  s } )
653ad2ant2 977 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  |^| { s  e.  C  |  U  C_  s }  C_  |^| { s  e.  C  |  V  C_  s } )
7 simp1 955 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
8 sstr 3187 . . . 4  |-  ( ( U  C_  V  /\  V  C_  X )  ->  U  C_  X )
983adant1 973 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  U  C_  X )
10 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1110mrcval 13512 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
127, 9, 11syl2anc 642 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
1310mrcval 13512 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
14133adant2 974 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
156, 12, 143sstr4d 3221 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   |^|cint 3862   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  mrcsscl  13522  mrcuni  13523  mrcssd  13526  isacs3lem  14269  isacs4lem  14271  dprdres  15263  dprdss  15264  dprd2dlem1  15276  dprd2da  15277  dmdprdsplit2lem  15280  ismrc  26776  isnacs3  26785
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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