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Theorem mrcss 13841
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 3355 . . . . . 6  |-  ( U 
C_  V  ->  ( V  C_  s  ->  U  C_  s ) )
21adantr 452 . . . . 5  |-  ( ( U  C_  V  /\  s  e.  C )  ->  ( V  C_  s  ->  U  C_  s )
)
32ss2rabdv 3424 . . . 4  |-  ( U 
C_  V  ->  { s  e.  C  |  V  C_  s }  C_  { s  e.  C  |  U  C_  s } )
4 intss 4071 . . . 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 979 . 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 957 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
8 sstr 3356 . . . 4  |-  ( ( U  C_  V  /\  V  C_  X )  ->  U  C_  X )
983adant1 975 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  U  C_  X )
10 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1110mrcval 13835 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
127, 9, 11syl2anc 643 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
1310mrcval 13835 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
14133adant2 976 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  V )  =  |^| { s  e.  C  |  V  C_  s } )
156, 12, 143sstr4d 3391 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 936    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   |^|cint 4050   ` cfv 5454  Moorecmre 13807  mrClscmrc 13808
This theorem is referenced by:  mrcsscl  13845  mrcuni  13846  mrcssd  13849  ismrc  26755  isnacs3  26764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-mre 13811  df-mrc 13812
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