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Theorem pclfvalN 39254
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 3485 . 2 (𝐾𝑉𝐾 ∈ V)
2 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
3 fveq2 6882 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pclfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2782 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65pweqd 4612 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7 fveq2 6882 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
97, 8eqtr4di 2782 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
109rabeqdv 3439 . . . . . 6 (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
1110inteqd 4946 . . . . 5 (𝑘 = 𝐾 {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
126, 11mpteq12dv 5230 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
13 df-pclN 39253 . . . 4 PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
144fvexi 6896 . . . . . 6 𝐴 ∈ V
1514pwex 5369 . . . . 5 𝒫 𝐴 ∈ V
1615mptex 7217 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) ∈ V
1712, 13, 16fvmpt 6989 . . 3 (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
182, 17eqtrid 2776 . 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 1533  wcel 2098  {crab 3424  Vcvv 3466  wss 3941  𝒫 cpw 4595   cint 4941  cmpt 5222  cfv 6534  Atomscatm 38627  PSubSpcpsubsp 38861  PClcpclN 39252
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-pclN 39253
This theorem is referenced by:  pclvalN  39255
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