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Theorem pclfvalN 37085
 Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a 𝐴 = (Atoms‘𝐾)
pclfval.s 𝑆 = (PSubSp‘𝐾)
pclfval.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclfvalN (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐾,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem pclfvalN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3497 . 2 (𝐾𝑉𝐾 ∈ V)
2 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
3 fveq2 6651 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pclfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2877 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65pweqd 4539 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7 fveq2 6651 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
97, 8syl6eqr 2877 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
109rabeqdv 3469 . . . . . 6 (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
1110inteqd 4862 . . . . 5 (𝑘 = 𝐾 {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
126, 11mpteq12dv 5132 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
13 df-pclN 37084 . . . 4 PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
144fvexi 6665 . . . . . 6 𝐴 ∈ V
1514pwex 5262 . . . . 5 𝒫 𝐴 ∈ V
1615mptex 6967 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) ∈ V
1712, 13, 16fvmpt 6749 . . 3 (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
182, 17syl5eq 2871 . 2 (𝐾 ∈ V → 𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
191, 18syl 17 1 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  {crab 3136  Vcvv 3479   ⊆ wss 3918  𝒫 cpw 4520  ∩ cint 4857   ↦ cmpt 5127  ‘cfv 6336  Atomscatm 36459  PSubSpcpsubsp 36692  PClcpclN 37083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-pclN 37084 This theorem is referenced by:  pclvalN  37086
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