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Theorem mrcval 13840
Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcval  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
Distinct variable groups:    F, s    C, s    X, s    U, s

Proof of Theorem mrcval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
21mrcfval 13838 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
32adantr 453 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  F  =  ( x  e. 
~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) )
4 sseq1 3371 . . . . 5  |-  ( x  =  U  ->  (
x  C_  s  <->  U  C_  s
) )
54rabbidv 2950 . . . 4  |-  ( x  =  U  ->  { s  e.  C  |  x 
C_  s }  =  { s  e.  C  |  U  C_  s } )
65inteqd 4057 . . 3  |-  ( x  =  U  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
76adantl 454 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  X )  /\  x  =  U )  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
8 mre1cl 13824 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
9 elpw2g 4366 . . . 4  |-  ( X  e.  C  ->  ( U  e.  ~P X  <->  U 
C_  X ) )
108, 9syl 16 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  ~P X  <->  U  C_  X
) )
1110biimpar 473 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
128adantr 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  C )
13 simpr 449 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  X )
14 sseq2 3372 . . . . . 6  |-  ( s  =  X  ->  ( U  C_  s  <->  U  C_  X
) )
1514elrab 3094 . . . . 5  |-  ( X  e.  { s  e.  C  |  U  C_  s }  <->  ( X  e.  C  /\  U  C_  X ) )
1612, 13, 15sylanbrc 647 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  { s  e.  C  |  U  C_  s } )
17 ne0i 3636 . . . 4  |-  ( X  e.  { s  e.  C  |  U  C_  s }  ->  { s  e.  C  |  U  C_  s }  =/=  (/) )
1816, 17syl 16 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  { s  e.  C  |  U  C_  s }  =/=  (/) )
19 intex 4359 . . 3  |-  ( { s  e.  C  |  U  C_  s }  =/=  (/)  <->  |^|
{ s  e.  C  |  U  C_  s }  e.  _V )
2018, 19sylib 190 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  |^| { s  e.  C  |  U  C_  s }  e.  _V )
213, 7, 11, 20fvmptd 5813 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052    e. cmpt 4269   ` cfv 5457  Moorecmre 13812  mrClscmrc 13813
This theorem is referenced by:  mrcid  13843  mrcss  13846  mrcssid  13847  cycsubg2  14982  aspval2  16410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-mre 13816  df-mrc 13817
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