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Theorem elpclN 36068
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 36066 . . 3  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
54eleq2d 2466 . 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 4228 . 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 367    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750   _Vcvv 3051    C_ wss 3406   |^|cint 4216   ` cfv 5513   Atomscatm 35440   PSubSpcpsubsp 35672   PClcpclN 36063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-psubsp 35679  df-pclN 36064
This theorem is referenced by:  pclfinclN  36126
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