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Theorem pclfvalN 39891
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 3501 . 2 (𝐾𝑉𝐾 ∈ V)
2 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
3 fveq2 6906 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pclfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2795 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65pweqd 4617 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
97, 8eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
109rabeqdv 3452 . . . . . 6 (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
1110inteqd 4951 . . . . 5 (𝑘 = 𝐾 {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
126, 11mpteq12dv 5233 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
13 df-pclN 39890 . . . 4 PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
144fvexi 6920 . . . . . 6 𝐴 ∈ V
1514pwex 5380 . . . . 5 𝒫 𝐴 ∈ V
1615mptex 7243 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) ∈ V
1712, 13, 16fvmpt 7016 . . 3 (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
182, 17eqtrid 2789 . 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 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   cint 4946  cmpt 5225  cfv 6561  Atomscatm 39264  PSubSpcpsubsp 39498  PClcpclN 39889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-pclN 39890
This theorem is referenced by:  pclvalN  39892
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