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Theorem pclfvalN 30078
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 pclfval.c . . 3  |-  U  =  ( PCl `  K
)
3 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pclfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65pweqd 3630 . . . . 5  |-  ( k  =  K  ->  ~P ( Atoms `  k )  =  ~P A )
7 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
8 pclfval.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
97, 8syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
10 biidd 228 . . . . . . 7  |-  ( k  =  K  ->  (
x  C_  y  <->  x  C_  y
) )
119, 10rabeqbidv 2783 . . . . . 6  |-  ( k  =  K  ->  { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  {
y  e.  S  |  x  C_  y } )
1211inteqd 3867 . . . . 5  |-  ( k  =  K  ->  |^| { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  |^| { y  e.  S  |  x  C_  y } )
136, 12mpteq12dv 4098 . . . 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 30077 . . . 4  |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
|^| { y  e.  (
PSubSp `  k )  |  x  C_  y }
) )
15 fvex 5539 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
164, 15eqeltri 2353 . . . . . 6  |-  A  e. 
_V
1716pwex 4193 . . . . 5  |-  ~P A  e.  _V
1817mptex 5746 . . . 4  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  e.  _V
1913, 14, 18fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( PCl `  K )  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) )
202, 19syl5eq 2327 . 2  |-  ( K  e.  _V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
211, 20syl 15 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862    e. cmpt 4077   ` cfv 5255   Atomscatm 29453   PSubSpcpsubsp 29685   PClcpclN 30076
This theorem is referenced by:  pclvalN  30079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-pclN 30077
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