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Theorem elpclN 39267
Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a 𝐴 = (Atomsβ€˜πΎ)
pclfval.s 𝑆 = (PSubSpβ€˜πΎ)
pclfval.c π‘ˆ = (PClβ€˜πΎ)
elpcl.q 𝑄 ∈ V
Assertion
Ref Expression
elpclN ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋   𝑦,𝑉   𝑦,𝑄
Allowed substitution hint:   π‘ˆ(𝑦)

Proof of Theorem elpclN
StepHypRef Expression
1 pclfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 pclfval.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 pclfval.c . . . 4 π‘ˆ = (PClβ€˜πΎ)
41, 2, 3pclvalN 39265 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
54eleq2d 2811 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))
6 elpcl.q . . 3 𝑄 ∈ V
76elintrab 4955 . 2 (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦))
85, 7bitrdi 287 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  {crab 3424  Vcvv 3466   βŠ† wss 3941  βˆ© cint 4941  β€˜cfv 6534  Atomscatm 38637  PSubSpcpsubsp 38871  PClcpclN 39262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-psubsp 38878  df-pclN 39263
This theorem is referenced by:  pclfinclN  39325
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