MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcval Structured version   Unicode version

Theorem mrcval 14668
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 14666 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
32adantr 465 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  F  =  ( x  e. 
~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) )
4 sseq1 3486 . . . . 5  |-  ( x  =  U  ->  (
x  C_  s  <->  U  C_  s
) )
54rabbidv 3070 . . . 4  |-  ( x  =  U  ->  { s  e.  C  |  x 
C_  s }  =  { s  e.  C  |  U  C_  s } )
65inteqd 4242 . . 3  |-  ( x  =  U  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
76adantl 466 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  X )  /\  x  =  U )  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
8 mre1cl 14652 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
9 elpw2g 4564 . . . 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 485 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
128adantr 465 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  C )
13 simpr 461 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  X )
14 sseq2 3487 . . . . . 6  |-  ( s  =  X  ->  ( U  C_  s  <->  U  C_  X
) )
1514elrab 3224 . . . . 5  |-  ( X  e.  { s  e.  C  |  U  C_  s }  <->  ( X  e.  C  /\  U  C_  X ) )
1612, 13, 15sylanbrc 664 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  { s  e.  C  |  U  C_  s } )
17 ne0i 3752 . . . 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 4557 . . 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 5889 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 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   |^|cint 4237    |-> cmpt 4459   ` cfv 5527  Moorecmre 14640  mrClscmrc 14641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-mre 14644  df-mrc 14645
This theorem is referenced by:  mrcid  14671  mrcss  14674  mrcssid  14675  cycsubg2  15838  aspval2  17541
  Copyright terms: Public domain W3C validator