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Theorem mrcss 13567
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 3220 . . . . . 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 3288 . . . 4  |-  ( U 
C_  V  ->  { s  e.  C  |  V  C_  s }  C_  { s  e.  C  |  U  C_  s } )
4 intss 3920 . . . 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 3221 . . . 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 13561 . . 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 13561 . . 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 3255 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 1633    e. wcel 1701   {crab 2581    C_ wss 3186   |^|cint 3899   ` cfv 5292  Moorecmre 13533  mrClscmrc 13534
This theorem is referenced by:  mrcsscl  13571  mrcuni  13572  mrcssd  13575  isacs3lem  14318  isacs4lem  14320  dprdres  15312  dprdss  15313  dprd2dlem1  15325  dprd2da  15326  dmdprdsplit2lem  15329  ismrc  25924  isnacs3  25933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-mre 13537  df-mrc 13538
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