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Theorem pclvalN 36507
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 6544 . . 3 𝐴 ∈ V
32elpw2 5132 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
4 pclfval.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
5 pclfval.c . . . . . 6 𝑈 = (PCl‘𝐾)
61, 4, 5pclfvalN 36506 . . . . 5 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
76fveq1d 6532 . . . 4 (𝐾𝑉 → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
87adantr 481 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
9 eqid 2793 . . . 4 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})
10 sseq1 3908 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
1110rabbidv 3420 . . . . 5 (𝑥 = 𝑋 → {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
1211inteqd 4781 . . . 4 (𝑥 = 𝑋 {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
13 simpr 485 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14 elpwi 4457 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
1514adantl 482 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋𝐴)
161, 4atpsubN 36370 . . . . . . . . 9 (𝐾𝑉𝐴𝑆)
1716adantr 481 . . . . . . . 8 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴𝑆)
18 sseq2 3909 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
1918elrab3 3614 . . . . . . . 8 (𝐴𝑆 → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2017, 19syl 17 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
2115, 20mpbird 258 . . . . . 6 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦𝑆𝑋𝑦})
2221ne0d 4215 . . . . 5 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ≠ ∅)
23 intex 5124 . . . . 5 ({𝑦𝑆𝑋𝑦} ≠ ∅ ↔ {𝑦𝑆𝑋𝑦} ∈ V)
2422, 23sylib 219 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ∈ V)
259, 12, 13, 24fvmptd3 6648 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
268, 25eqtrd 2829 . 2 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
273, 26sylan2br 594 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1520  wcel 2079  wne 2982  {crab 3107  Vcvv 3432  wss 3854  c0 4206  𝒫 cpw 4447   cint 4776  cmpt 5035  cfv 6217  Atomscatm 35880  PSubSpcpsubsp 36113  PClcpclN 36504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-psubsp 36120  df-pclN 36505
This theorem is referenced by:  pclclN  36508  elpclN  36509  elpcliN  36510  pclssN  36511  pclssidN  36512  pclidN  36513
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