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Theorem pclfvalN 30623
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 2956 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 pclfval.c . . 3  |-  U  =  ( PCl `  K
)
3 fveq2 5720 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pclfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65pweqd 3796 . . . . 5  |-  ( k  =  K  ->  ~P ( Atoms `  k )  =  ~P A )
7 fveq2 5720 . . . . . . . 8  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  ( PSubSp `  K )
)
8 pclfval.s . . . . . . . 8  |-  S  =  ( PSubSp `  K )
97, 8syl6eqr 2485 . . . . . . 7  |-  ( k  =  K  ->  ( PSubSp `
 k )  =  S )
10 biidd 229 . . . . . . 7  |-  ( k  =  K  ->  (
x  C_  y  <->  x  C_  y
) )
119, 10rabeqbidv 2943 . . . . . 6  |-  ( k  =  K  ->  { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  {
y  e.  S  |  x  C_  y } )
1211inteqd 4047 . . . . 5  |-  ( k  =  K  ->  |^| { y  e.  ( PSubSp `  k
)  |  x  C_  y }  =  |^| { y  e.  S  |  x  C_  y } )
136, 12mpteq12dv 4279 . . . 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 30622 . . . 4  |-  PCl  =  ( k  e.  _V  |->  ( x  e.  ~P ( Atoms `  k )  |-> 
|^| { y  e.  (
PSubSp `  k )  |  x  C_  y }
) )
15 fvex 5734 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
164, 15eqeltri 2505 . . . . . 6  |-  A  e. 
_V
1716pwex 4374 . . . . 5  |-  ~P A  e.  _V
1817mptex 5958 . . . 4  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  e.  _V
1913, 14, 18fvmpt 5798 . . 3  |-  ( K  e.  _V  ->  ( PCl `  K )  =  ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) )
202, 19syl5eq 2479 . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   |^|cint 4042    e. cmpt 4258   ` cfv 5446   Atomscatm 29998   PSubSpcpsubsp 30230   PClcpclN 30621
This theorem is referenced by:  pclvalN  30624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-pclN 30622
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