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Theorem pclfvalN 35698
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 3364 . 2 (𝐾𝑉𝐾 ∈ V)
2 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
3 fveq2 6333 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pclfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2823 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65pweqd 4303 . . . . 5 (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7 fveq2 6333 . . . . . . . 8 (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSp‘𝐾)
97, 8syl6eqr 2823 . . . . . . 7 (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
109rabeqdv 3344 . . . . . 6 (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
1110inteqd 4617 . . . . 5 (𝑘 = 𝐾 {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦} = {𝑦𝑆𝑥𝑦})
126, 11mpteq12dv 4868 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
13 df-pclN 35697 . . . 4 PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
144fvexi 6345 . . . . . 6 𝐴 ∈ V
1514pwex 4982 . . . . 5 𝒫 𝐴 ∈ V
1615mptex 6633 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) ∈ V
1712, 13, 16fvmpt 6426 . . 3 (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
182, 17syl5eq 2817 . 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 1631  wcel 2145  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4298   cint 4612  cmpt 4864  cfv 6030  Atomscatm 35072  PSubSpcpsubsp 35305  PClcpclN 35696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-pclN 35697
This theorem is referenced by:  pclvalN  35699
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