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Theorem pclfvalN 30003
Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a  |-  A  =  ( Atoms `  K )
pclfval.s  |-  S  =  ( PSubSp `  K )
pclfval.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclfvalN  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
Distinct variable groups:    x, y, A    x, K, y    x, S, y
Allowed substitution hints:    U( x, y)    V( x, y)

Proof of Theorem pclfvalN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2907 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 pclfval.c . . 3  |-  U  =  ( PCl `  K
)
3 fveq2 5668 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pclfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2437 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65pweqd 3747 . . . . 5  |-  ( k  =  K  ->  ~P ( Atoms `  k )  =  ~P A )
7 fveq2 5668 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
8 pclfval.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
97, 8syl6eqr 2437 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
10 biidd 229 . . . . . . 7  |-  ( k  =  K  ->  (
x  C_  y  <->  x  C_  y
) )
119, 10rabeqbidv 2894 . . . . . 6  |-  ( k  =  K  ->  { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  {
y  e.  S  |  x  C_  y } )
1211inteqd 3997 . . . . 5  |-  ( k  =  K  ->  |^| { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  |^| { y  e.  S  |  x  C_  y } )
136, 12mpteq12dv 4228 . . . 4  |-  ( k  =  K  ->  (
x  e.  ~P ( Atoms `  k )  |->  |^|
{ y  e.  (
PSubSp `  k )  |  x  C_  y }
)  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) )
14 df-pclN 30002 . . . 4  |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
|^| { y  e.  (
PSubSp `  k )  |  x  C_  y }
) )
15 fvex 5682 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
164, 15eqeltri 2457 . . . . . 6  |-  A  e. 
_V
1716pwex 4323 . . . . 5  |-  ~P A  e.  _V
1817mptex 5905 . . . 4  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  e.  _V
1913, 14, 18fvmpt 5745 . . 3  |-  ( K  e.  _V  ->  ( PCl `  K )  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) )
202, 19syl5eq 2431 . 2  |-  ( K  e.  _V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
211, 20syl 16 1  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   |^|cint 3992    e. cmpt 4207   ` cfv 5394   Atomscatm 29378   PSubSpcpsubsp 29610   PClcpclN 30001
This theorem is referenced by:  pclvalN  30004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-pclN 30002
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