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Theorem elpclN 39874
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 ∈ V
Assertion
Ref Expression
elpclN ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))
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 39872 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
54eleq2d 2824 . 2 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑦𝑆𝑋𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4964 . 2 (𝑄 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦))
85, 7bitrdi 287 1 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  wss 3962   cint 4950  cfv 6562  Atomscatm 39244  PSubSpcpsubsp 39478  PClcpclN 39869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-psubsp 39485  df-pclN 39870
This theorem is referenced by:  pclfinclN  39932
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