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Theorem pclvalN 37337
 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 6669 . . 3 𝐴 ∈ V
32elpw2 5216 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
4 pclfval.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
5 pclfval.c . . . . . 6 𝑈 = (PCl‘𝐾)
61, 4, 5pclfvalN 37336 . . . . 5 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
76fveq1d 6657 . . . 4 (𝐾𝑉 → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
87adantr 484 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
9 eqid 2798 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})
10 sseq1 3942 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
1110rabbidv 3428 . . . . 5 (𝑥 = 𝑋 → {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
1211inteqd 4847 . . . 4 (𝑥 = 𝑋 {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
13 simpr 488 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14 elpwi 4509 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
1514adantl 485 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋𝐴)
161, 4atpsubN 37200 . . . . . . . . 9 (𝐾𝑉𝐴𝑆)
1716adantr 484 . . . . . . . 8 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴𝑆)
18 sseq2 3943 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
1918elrab3 3631 . . . . . . . 8 (𝐴𝑆 → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2017, 19syl 17 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2115, 20mpbird 260 . . . . . 6 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦𝑆𝑋𝑦})
2221ne0d 4254 . . . . 5 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ≠ ∅)
23 intex 5208 . . . . 5 ({𝑦𝑆𝑋𝑦} ≠ ∅ ↔ {𝑦𝑆𝑋𝑦} ∈ V)
2422, 23sylib 221 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ∈ V)
259, 12, 13, 24fvmptd3 6778 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
268, 25eqtrd 2833 . 2 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
273, 26sylan2br 597 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3442   ⊆ wss 3883  ∅c0 4246  𝒫 cpw 4500  ∩ cint 4842   ↦ cmpt 5114  ‘cfv 6332  Atomscatm 36710  PSubSpcpsubsp 36943  PClcpclN 37334 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-psubsp 36950  df-pclN 37335 This theorem is referenced by:  pclclN  37338  elpclN  37339  elpcliN  37340  pclssN  37341  pclssidN  37342  pclidN  37343
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