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Theorem pclvalN 30055
Description: Value of 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
pclvalN  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
Distinct variable groups:    y, A    y, K    y, S    y, X
Allowed substitution hints:    U( y)    V( y)

Proof of Theorem pclvalN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . . 4  |-  A  =  ( Atoms `  K )
2 fvex 5675 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2450 . . 3  |-  A  e. 
_V
43elpw2 4298 . 2  |-  ( X  e.  ~P A  <->  X  C_  A
)
5 pclfval.s . . . . . 6  |-  S  =  ( PSubSp `  K )
6 pclfval.c . . . . . 6  |-  U  =  ( PCl `  K
)
71, 5, 6pclfvalN 30054 . . . . 5  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
87fveq1d 5663 . . . 4  |-  ( K  e.  V  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X ) )
98adantr 452 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) `  X
) )
10 simpr 448 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  e.  ~P A )
11 elpwi 3743 . . . . . . . 8  |-  ( X  e.  ~P A  ->  X  C_  A )
1211adantl 453 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  C_  A
)
131, 5atpsubN 29918 . . . . . . . . 9  |-  ( K  e.  V  ->  A  e.  S )
1413adantr 452 . . . . . . . 8  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  S )
15 sseq2 3306 . . . . . . . . 9  |-  ( y  =  A  ->  ( X  C_  y  <->  X  C_  A
) )
1615elrab3 3029 . . . . . . . 8  |-  ( A  e.  S  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <->  X  C_  A
) )
1714, 16syl 16 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <-> 
X  C_  A )
)
1812, 17mpbird 224 . . . . . 6  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  { y  e.  S  |  X  C_  y } )
19 ne0i 3570 . . . . . 6  |-  ( A  e.  { y  e.  S  |  X  C_  y }  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
2018, 19syl 16 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
21 intex 4290 . . . . 5  |-  ( { y  e.  S  |  X  C_  y }  =/=  (/)  <->  |^|
{ y  e.  S  |  X  C_  y }  e.  _V )
2220, 21sylib 189 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  |^| { y  e.  S  |  X  C_  y }  e.  _V )
23 sseq1 3305 . . . . . . 7  |-  ( x  =  X  ->  (
x  C_  y  <->  X  C_  y
) )
2423rabbidv 2884 . . . . . 6  |-  ( x  =  X  ->  { y  e.  S  |  x 
C_  y }  =  { y  e.  S  |  X  C_  y } )
2524inteqd 3990 . . . . 5  |-  ( x  =  X  ->  |^| { y  e.  S  |  x 
C_  y }  =  |^| { y  e.  S  |  X  C_  y } )
26 eqid 2380 . . . . 5  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )
2725, 26fvmptg 5736 . . . 4  |-  ( ( X  e.  ~P A  /\  |^| { y  e.  S  |  X  C_  y }  e.  _V )  ->  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X )  =  |^| { y  e.  S  |  X  C_  y } )
2810, 22, 27syl2anc 643 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( (
x  e.  ~P A  |-> 
|^| { y  e.  S  |  x  C_  y } ) `  X )  =  |^| { y  e.  S  |  X  C_  y } )
299, 28eqtrd 2412 . 2  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
304, 29sylan2br 463 1  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   {crab 2646   _Vcvv 2892    C_ wss 3256   (/)c0 3564   ~Pcpw 3735   |^|cint 3985    e. cmpt 4200   ` cfv 5387   Atomscatm 29429   PSubSpcpsubsp 29661   PClcpclN 30052
This theorem is referenced by:  pclclN  30056  elpclN  30057  elpcliN  30058  pclssN  30059  pclssidN  30060  pclidN  30061
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-psubsp 29668  df-pclN 30053
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