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Theorem mrcss 14674
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 3472 . . . . . 6  |-  ( U 
C_  V  ->  ( V  C_  s  ->  U  C_  s ) )
21adantr 465 . . . . 5  |-  ( ( U  C_  V  /\  s  e.  C )  ->  ( V  C_  s  ->  U  C_  s )
)
32ss2rabdv 3542 . . . 4  |-  ( U 
C_  V  ->  { s  e.  C  |  V  C_  s }  C_  { s  e.  C  |  U  C_  s } )
4 intss 4258 . . . 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 16 . . 3  |-  ( U 
C_  V  ->  |^| { s  e.  C  |  U  C_  s }  C_  |^| { s  e.  C  |  V  C_  s } )
653ad2ant2 1010 . 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 988 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
8 sstr 3473 . . . 4  |-  ( ( U  C_  V  /\  V  C_  X )  ->  U  C_  X )
983adant1 1006 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  U  C_  X )
10 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1110mrcval 14668 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
127, 9, 11syl2anc 661 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
1310mrcval 14668 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
14133adant2 1007 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
156, 12, 143sstr4d 3508 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3437   |^|cint 4237   ` cfv 5527  Moorecmre 14640  mrClscmrc 14641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-mre 14644  df-mrc 14645
This theorem is referenced by:  mrcsscl  14678  mrcuni  14679  mrcssd  14682  ismrc  29186  isnacs3  29195
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