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Theorem pclfvalN 38355
Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a 𝐴 = (Atomsβ€˜πΎ)
pclfval.s 𝑆 = (PSubSpβ€˜πΎ)
pclfval.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclfvalN (𝐾 ∈ 𝑉 β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐾,𝑦   π‘₯,𝑆,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝑉(π‘₯,𝑦)

Proof of Theorem pclfvalN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3464 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 pclfval.c . . 3 π‘ˆ = (PClβ€˜πΎ)
3 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 pclfval.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2795 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65pweqd 4578 . . . . 5 (π‘˜ = 𝐾 β†’ 𝒫 (Atomsβ€˜π‘˜) = 𝒫 𝐴)
7 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (PSubSpβ€˜π‘˜) = (PSubSpβ€˜πΎ))
8 pclfval.s . . . . . . . 8 𝑆 = (PSubSpβ€˜πΎ)
97, 8eqtr4di 2795 . . . . . . 7 (π‘˜ = 𝐾 β†’ (PSubSpβ€˜π‘˜) = 𝑆)
109rabeqdv 3423 . . . . . 6 (π‘˜ = 𝐾 β†’ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})
1110inteqd 4913 . . . . 5 (π‘˜ = 𝐾 β†’ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})
126, 11mpteq12dv 5197 . . . 4 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}) = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
13 df-pclN 38354 . . . 4 PCl = (π‘˜ ∈ V ↦ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}))
144fvexi 6857 . . . . . 6 𝐴 ∈ V
1514pwex 5336 . . . . 5 𝒫 𝐴 ∈ V
1615mptex 7174 . . . 4 (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}) ∈ V
1712, 13, 16fvmpt 6949 . . 3 (𝐾 ∈ V β†’ (PClβ€˜πΎ) = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
182, 17eqtrid 2789 . 2 (𝐾 ∈ V β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
191, 18syl 17 1 (𝐾 ∈ 𝑉 β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3408  Vcvv 3446   βŠ† wss 3911  π’« cpw 4561  βˆ© cint 4908   ↦ cmpt 5189  β€˜cfv 6497  Atomscatm 37728  PSubSpcpsubsp 37962  PClcpclN 38353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-pclN 38354
This theorem is referenced by:  pclvalN  38356
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