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Theorem pclvalN 38356
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 6857 . . 3 𝐴 ∈ V
32elpw2 5303 . 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 38355 . . . . 5 (𝐾 ∈ 𝑉 β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
76fveq1d 6845 . . . 4 (𝐾 ∈ 𝑉 β†’ (π‘ˆβ€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹))
87adantr 482 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹))
9 eqid 2737 . . . 4 (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}) = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})
10 sseq1 3970 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑦 ↔ 𝑋 βŠ† 𝑦))
1110rabbidv 3416 . . . . 5 (π‘₯ = 𝑋 β†’ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
1211inteqd 4913 . . . 4 (π‘₯ = 𝑋 β†’ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
13 simpr 486 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝑋 ∈ 𝒫 𝐴)
14 elpwi 4568 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴 β†’ 𝑋 βŠ† 𝐴)
1514adantl 483 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝑋 βŠ† 𝐴)
161, 4atpsubN 38219 . . . . . . . . 9 (𝐾 ∈ 𝑉 β†’ 𝐴 ∈ 𝑆)
1716adantr 482 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝐴 ∈ 𝑆)
18 sseq2 3971 . . . . . . . . 9 (𝑦 = 𝐴 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝐴))
1918elrab3 3647 . . . . . . . 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 257 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
2221ne0d 4296 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} β‰  βˆ…)
23 intex 5295 . . . . 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 6972 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ ((π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦})β€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
268, 25eqtrd 2777 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
273, 26sylan2br 596 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3408  Vcvv 3446   βŠ† wss 3911  βˆ…c0 4283  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-ov 7361  df-psubsp 37969  df-pclN 38354
This theorem is referenced by:  pclclN  38357  elpclN  38358  elpcliN  38359  pclssN  38360  pclssidN  38361  pclidN  38362
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