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Theorem pclvalN 39395
Description: Value of 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
pclvalN ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋
Allowed substitution hints:   π‘ˆ(𝑦)   𝑉(𝑦)

Proof of Theorem pclvalN
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
21fvexi 6916 . . 3 𝐴 ∈ V
32elpw2 5351 . 2 (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 βŠ† 𝐴)
4 pclfval.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
5 pclfval.c . . . . . 6 π‘ˆ = (PClβ€˜πΎ)
61, 4, 5pclfvalN 39394 . . . . 5 (𝐾 ∈ 𝑉 β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
76fveq1d 6904 . . . 4 (𝐾 ∈ 𝑉 β†’ (π‘ˆβ€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹))
87adantr 479 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹))
9 eqid 2728 . . . 4 (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}) = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})
10 sseq1 4007 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑦 ↔ 𝑋 βŠ† 𝑦))
1110rabbidv 3438 . . . . 5 (π‘₯ = 𝑋 β†’ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
1211inteqd 4958 . . . 4 (π‘₯ = 𝑋 β†’ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
13 simpr 483 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝑋 ∈ 𝒫 𝐴)
14 elpwi 4613 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴 β†’ 𝑋 βŠ† 𝐴)
1514adantl 480 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝑋 βŠ† 𝐴)
161, 4atpsubN 39258 . . . . . . . . 9 (𝐾 ∈ 𝑉 β†’ 𝐴 ∈ 𝑆)
1716adantr 479 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝐴 ∈ 𝑆)
18 sseq2 4008 . . . . . . . . 9 (𝑦 = 𝐴 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝐴))
1918elrab3 3685 . . . . . . . 8 (𝐴 ∈ 𝑆 β†’ (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ 𝑋 βŠ† 𝐴))
2017, 19syl 17 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ 𝑋 βŠ† 𝐴))
2115, 20mpbird 256 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
2221ne0d 4339 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} β‰  βˆ…)
23 intex 5343 . . . . 5 ({𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} β‰  βˆ… ↔ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ V)
2422, 23sylib 217 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ V)
259, 12, 13, 24fvmptd3 7033 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
268, 25eqtrd 2768 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
273, 26sylan2br 593 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  {crab 3430  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  βˆ© cint 4953   ↦ cmpt 5235  β€˜cfv 6553  Atomscatm 38767  PSubSpcpsubsp 39001  PClcpclN 39392
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-psubsp 39008  df-pclN 39393
This theorem is referenced by:  pclclN  39396  elpclN  39397  elpcliN  39398  pclssN  39399  pclssidN  39400  pclidN  39401
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