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Theorem elpclN 37027
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 37025 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
54eleq2d 2898 . 2 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑦𝑆𝑋𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4887 . 2 (𝑄 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦))
85, 7syl6bb 289 1 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  {crab 3142  Vcvv 3494  wss 3935   cint 4875  cfv 6354  Atomscatm 36398  PSubSpcpsubsp 36631  PClcpclN 37022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-psubsp 36638  df-pclN 37023
This theorem is referenced by:  pclfinclN  37085
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