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Theorem elpclN 33875
Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a  |-  A  =  ( Atoms `  K )
pclfval.s  |-  S  =  ( PSubSp `  K )
pclfval.c  |-  U  =  ( PCl `  K
)
elpcl.q  |-  Q  e. 
_V
Assertion
Ref Expression
elpclN  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( Q  e.  ( U `  X )  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
) )
Distinct variable groups:    y, A    y, K    y, S    y, X    y, V    y, Q
Allowed substitution hint:    U( y)

Proof of Theorem elpclN
StepHypRef Expression
1 pclfval.a . . . 4  |-  A  =  ( Atoms `  K )
2 pclfval.s . . . 4  |-  S  =  ( PSubSp `  K )
3 pclfval.c . . . 4  |-  U  =  ( PCl `  K
)
41, 2, 3pclvalN 33873 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
54eleq2d 2524 . 2  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( Q  e.  ( U `  X )  <-> 
Q  e.  |^| { y  e.  S  |  X  C_  y } ) )
6 elpcl.q . . 3  |-  Q  e. 
_V
76elintrab 4249 . 2  |-  ( Q  e.  |^| { y  e.  S  |  X  C_  y }  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
)
85, 7syl6bb 261 1  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( Q  e.  ( U `  X )  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078    C_ wss 3437   |^|cint 4237   ` cfv 5527   Atomscatm 33247   PSubSpcpsubsp 33479   PClcpclN 33870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-psubsp 33486  df-pclN 33871
This theorem is referenced by:  pclfinclN  33933
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