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Theorem pclfvalN 39872
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 3499 . 2 (𝐾𝑉𝐾 ∈ V)
2 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
3 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pclfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2793 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65pweqd 4622 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
97, 8eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
109rabeqdv 3449 . . . . . 6 (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
1110inteqd 4956 . . . . 5 (𝑘 = 𝐾 {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
126, 11mpteq12dv 5239 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
13 df-pclN 39871 . . . 4 PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
144fvexi 6921 . . . . . 6 𝐴 ∈ V
1514pwex 5386 . . . . 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 2787 . 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 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   cint 4951  cmpt 5231  cfv 6563  Atomscatm 39245  PSubSpcpsubsp 39479  PClcpclN 39870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-pclN 39871
This theorem is referenced by:  pclvalN  39873
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