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Theorem mrcss 15521
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 3471 . . . . . 6  |-  ( U 
C_  V  ->  ( V  C_  s  ->  U  C_  s ) )
21adantr 466 . . . . 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 4276 . . . 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 17 . . 3  |-  ( U 
C_  V  ->  |^| { s  e.  C  |  U  C_  s }  C_  |^| { s  e.  C  |  V  C_  s } )
653ad2ant2 1027 . 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 1005 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
8 sstr 3472 . . . 4  |-  ( ( U  C_  V  /\  V  C_  X )  ->  U  C_  X )
983adant1 1023 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  U  C_  X )
10 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1110mrcval 15515 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
127, 9, 11syl2anc 665 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
1310mrcval 15515 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
14133adant2 1024 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
156, 12, 143sstr4d 3507 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 982    = wceq 1437    e. wcel 1872   {crab 2775    C_ wss 3436   |^|cint 4255   ` cfv 5601  Moorecmre 15487  mrClscmrc 15488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-mre 15491  df-mrc 15492
This theorem is referenced by:  mrcsscl  15525  mrcuni  15526  mrcssd  15529  ismrc  35512  isnacs3  35521
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