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Theorem mrcval 13827
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 13825 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
32adantr 452 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  F  =  ( x  e. 
~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) )
4 sseq1 3361 . . . . 5  |-  ( x  =  U  ->  (
x  C_  s  <->  U  C_  s
) )
54rabbidv 2940 . . . 4  |-  ( x  =  U  ->  { s  e.  C  |  x 
C_  s }  =  { s  e.  C  |  U  C_  s } )
65inteqd 4047 . . 3  |-  ( x  =  U  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
76adantl 453 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  X )  /\  x  =  U )  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
8 mre1cl 13811 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
9 elpw2g 4355 . . . 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 472 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
128adantr 452 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  C )
13 simpr 448 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  X )
14 sseq2 3362 . . . . . 6  |-  ( s  =  X  ->  ( U  C_  s  <->  U  C_  X
) )
1514elrab 3084 . . . . 5  |-  ( X  e.  { s  e.  C  |  U  C_  s }  <->  ( X  e.  C  /\  U  C_  X ) )
1612, 13, 15sylanbrc 646 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  { s  e.  C  |  U  C_  s } )
17 ne0i 3626 . . . 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 4348 . . 3  |-  ( { s  e.  C  |  U  C_  s }  =/=  (/)  <->  |^|
{ s  e.  C  |  U  C_  s }  e.  _V )
2018, 19sylib 189 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  |^| { s  e.  C  |  U  C_  s }  e.  _V )
213, 7, 11, 20fvmptd 5802 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042    e. cmpt 4258   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800
This theorem is referenced by:  mrcid  13830  mrcss  13833  mrcssid  13834  cycsubg2  14969  aspval2  16397
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-mre 13803  df-mrc 13804
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