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Theorem elpclN 30689
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 30687 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
54eleq2d 2503 . 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 4062 . 2  |-  ( Q  e.  |^| { y  e.  S  |  X  C_  y }  <->  A. y  e.  S  ( X  C_  y  ->  Q  e.  y )
)
85, 7syl6bb 253 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    C_ wss 3320   |^|cint 4050   ` cfv 5454   Atomscatm 30061   PSubSpcpsubsp 30293   PClcpclN 30684
This theorem is referenced by:  pclfinclN  30747
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-psubsp 30300  df-pclN 30685
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