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Theorem pclfvalN 30700
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 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 pclfval.c . . 3  |-  U  =  ( PCl `  K
)
3 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pclfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65pweqd 3643 . . . . 5  |-  ( k  =  K  ->  ~P ( Atoms `  k )  =  ~P A )
7 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
8 pclfval.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
97, 8syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
10 biidd 228 . . . . . . 7  |-  ( k  =  K  ->  (
x  C_  y  <->  x  C_  y
) )
119, 10rabeqbidv 2796 . . . . . 6  |-  ( k  =  K  ->  { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  {
y  e.  S  |  x  C_  y } )
1211inteqd 3883 . . . . 5  |-  ( k  =  K  ->  |^| { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  |^| { y  e.  S  |  x  C_  y } )
136, 12mpteq12dv 4114 . . . 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 30699 . . . 4  |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
|^| { y  e.  (
PSubSp `  k )  |  x  C_  y }
) )
15 fvex 5555 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
164, 15eqeltri 2366 . . . . . 6  |-  A  e. 
_V
1716pwex 4209 . . . . 5  |-  ~P A  e.  _V
1817mptex 5762 . . . 4  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  e.  _V
1913, 14, 18fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( PCl `  K )  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) )
202, 19syl5eq 2340 . 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   ` cfv 5271   Atomscatm 30075   PSubSpcpsubsp 30307   PClcpclN 30698
This theorem is referenced by:  pclvalN  30701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pclN 30699
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