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Theorem mrcval 15594
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 15592 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
32adantr 472 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  F  =  ( x  e. 
~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) )
4 sseq1 3439 . . . . 5  |-  ( x  =  U  ->  (
x  C_  s  <->  U  C_  s
) )
54rabbidv 3022 . . . 4  |-  ( x  =  U  ->  { s  e.  C  |  x 
C_  s }  =  { s  e.  C  |  U  C_  s } )
65inteqd 4231 . . 3  |-  ( x  =  U  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
76adantl 473 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  X )  /\  x  =  U )  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  U  C_  s } )
8 mre1cl 15578 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
9 elpw2g 4564 . . . 4  |-  ( X  e.  C  ->  ( U  e.  ~P X  <->  U 
C_  X ) )
108, 9syl 17 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  ~P X  <->  U  C_  X
) )
1110biimpar 493 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
128adantr 472 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  C )
13 simpr 468 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  X )
14 sseq2 3440 . . . . . 6  |-  ( s  =  X  ->  ( U  C_  s  <->  U  C_  X
) )
1514elrab 3184 . . . . 5  |-  ( X  e.  { s  e.  C  |  U  C_  s }  <->  ( X  e.  C  /\  U  C_  X ) )
1612, 13, 15sylanbrc 677 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  X  e.  { s  e.  C  |  U  C_  s } )
17 ne0i 3728 . . . 4  |-  ( X  e.  { s  e.  C  |  U  C_  s }  ->  { s  e.  C  |  U  C_  s }  =/=  (/) )
1816, 17syl 17 . . 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 201 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  |^| { s  e.  C  |  U  C_  s }  e.  _V )
213, 7, 11, 20fvmptd 5969 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   |^|cint 4226    |-> cmpt 4454   ` cfv 5589  Moorecmre 15566  mrClscmrc 15567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-mre 15570  df-mrc 15571
This theorem is referenced by:  mrcid  15597  mrcss  15600  mrcssid  15601  cycsubg2  16932  aspval2  18648
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