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Theorem mrcval 15099
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 15097 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
32adantr 463 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  F  =  ( x  e. 
~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) )
4 sseq1 3510 . . . . 5  |-  ( x  =  U  ->  (
x  C_  s  <->  U  C_  s
) )
54rabbidv 3098 . . . 4  |-  ( x  =  U  ->  { s  e.  C  |  x 
C_  s }  =  { s  e.  C  |  U  C_  s } )
65inteqd 4276 . . 3  |-  ( x  =  U  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
76adantl 464 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  X )  /\  x  =  U )  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
8 mre1cl 15083 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
9 elpw2g 4600 . . . 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 483 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
128adantr 463 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  C )
13 simpr 459 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  X )
14 sseq2 3511 . . . . . 6  |-  ( s  =  X  ->  ( U  C_  s  <->  U  C_  X
) )
1514elrab 3254 . . . . 5  |-  ( X  e.  { s  e.  C  |  U  C_  s }  <->  ( X  e.  C  /\  U  C_  X ) )
1612, 13, 15sylanbrc 662 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  { s  e.  C  |  U  C_  s } )
17 ne0i 3789 . . . 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 4593 . . 3  |-  ( { s  e.  C  |  U  C_  s }  =/=  (/)  <->  |^|
{ s  e.  C  |  U  C_  s }  e.  _V )
2018, 19sylib 196 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  |^| { s  e.  C  |  U  C_  s }  e.  _V )
213, 7, 11, 20fvmptd 5936 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   |^|cint 4271    |-> cmpt 4497   ` cfv 5570  Moorecmre 15071  mrClscmrc 15072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-mre 15075  df-mrc 15076
This theorem is referenced by:  mrcid  15102  mrcss  15105  mrcssid  15106  cycsubg2  16437  aspval2  18191
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