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Theorem elpclN 35496
 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 35494 . . 3 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
54eleq2d 2716 . 2 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ 𝑄 {𝑦𝑆𝑋𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4520 . 2 (𝑄 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦))
85, 7syl6bb 276 1 ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∩ cint 4507  ‘cfv 5926  Atomscatm 34868  PSubSpcpsubsp 35100  PClcpclN 35491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-psubsp 35107  df-pclN 35492 This theorem is referenced by:  pclfinclN  35554
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