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Theorem elpclN 37914
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 37912 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
54eleq2d 2824 . 2 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑦𝑆𝑋𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4891 . 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3429  wss 3886   cint 4879  cfv 6426  Atomscatm 37285  PSubSpcpsubsp 37518  PClcpclN 37909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-psubsp 37525  df-pclN 37910
This theorem is referenced by:  pclfinclN  37972
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