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Theorem pclvalN 38566
Description: Value of 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
pclvalN ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋
Allowed substitution hints:   𝑈(𝑦)   𝑉(𝑦)

Proof of Theorem pclvalN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . . 4 𝐴 = (Atoms‘𝐾)
21fvexi 6892 . . 3 𝐴 ∈ V
32elpw2 5338 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
4 pclfval.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
5 pclfval.c . . . . . 6 𝑈 = (PCl‘𝐾)
61, 4, 5pclfvalN 38565 . . . . 5 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
76fveq1d 6880 . . . 4 (𝐾𝑉 → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
87adantr 481 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
9 eqid 2731 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})
10 sseq1 4003 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
1110rabbidv 3439 . . . . 5 (𝑥 = 𝑋 → {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
1211inteqd 4948 . . . 4 (𝑥 = 𝑋 {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
13 simpr 485 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14 elpwi 4603 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
1514adantl 482 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋𝐴)
161, 4atpsubN 38429 . . . . . . . . 9 (𝐾𝑉𝐴𝑆)
1716adantr 481 . . . . . . . 8 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴𝑆)
18 sseq2 4004 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
1918elrab3 3680 . . . . . . . 8 (𝐴𝑆 → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2017, 19syl 17 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2115, 20mpbird 256 . . . . . 6 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦𝑆𝑋𝑦})
2221ne0d 4331 . . . . 5 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ≠ ∅)
23 intex 5330 . . . . 5 ({𝑦𝑆𝑋𝑦} ≠ ∅ ↔ {𝑦𝑆𝑋𝑦} ∈ V)
2422, 23sylib 217 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ∈ V)
259, 12, 13, 24fvmptd3 7007 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
268, 25eqtrd 2771 . 2 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
273, 26sylan2br 595 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2939  {crab 3431  Vcvv 3473  wss 3944  c0 4318  𝒫 cpw 4596   cint 4943  cmpt 5224  cfv 6532  Atomscatm 37938  PSubSpcpsubsp 38172  PClcpclN 38563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-psubsp 38179  df-pclN 38564
This theorem is referenced by:  pclclN  38567  elpclN  38568  elpcliN  38569  pclssN  38570  pclssidN  38571  pclidN  38572
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