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Theorem pclfvalN 33533
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 2981 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 pclfval.c . . 3  |-  U  =  ( PCl `  K
)
3 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pclfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65pweqd 3865 . . . . 5  |-  ( k  =  K  ->  ~P ( Atoms `  k )  =  ~P A )
7 fveq2 5691 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
8 pclfval.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
97, 8syl6eqr 2493 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
10 biidd 237 . . . . . . 7  |-  ( k  =  K  ->  (
x  C_  y  <->  x  C_  y
) )
119, 10rabeqbidv 2967 . . . . . 6  |-  ( k  =  K  ->  { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  {
y  e.  S  |  x  C_  y } )
1211inteqd 4133 . . . . 5  |-  ( k  =  K  ->  |^| { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  |^| { y  e.  S  |  x  C_  y } )
136, 12mpteq12dv 4370 . . . 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 33532 . . . 4  |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
|^| { y  e.  (
PSubSp `  k )  |  x  C_  y }
) )
15 fvex 5701 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
164, 15eqeltri 2513 . . . . . 6  |-  A  e. 
_V
1716pwex 4475 . . . . 5  |-  ~P A  e.  _V
1817mptex 5948 . . . 4  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  e.  _V
1913, 14, 18fvmpt 5774 . . 3  |-  ( K  e.  _V  ->  ( PCl `  K )  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) )
202, 19syl5eq 2487 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   |^|cint 4128    e. cmpt 4350   ` cfv 5418   Atomscatm 32908   PSubSpcpsubsp 33140   PClcpclN 33531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-pclN 33532
This theorem is referenced by:  pclvalN  33534
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