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Theorem pclvalN 33977
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)
2 fvex 6097 . . . 4 (Atoms‘𝐾) ∈ V
31, 2eqeltri 2683 . . 3 𝐴 ∈ V
43elpw2 4749 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
5 pclfval.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
6 pclfval.c . . . . . 6 𝑈 = (PCl‘𝐾)
71, 5, 6pclfvalN 33976 . . . . 5 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
87fveq1d 6089 . . . 4 (𝐾𝑉 → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
98adantr 479 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
10 simpr 475 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
11 elpwi 4116 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
1211adantl 480 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋𝐴)
131, 5atpsubN 33840 . . . . . . . . 9 (𝐾𝑉𝐴𝑆)
1413adantr 479 . . . . . . . 8 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴𝑆)
15 sseq2 3589 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
1615elrab3 3331 . . . . . . . 8 (𝐴𝑆 → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
1714, 16syl 17 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
1812, 17mpbird 245 . . . . . 6 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦𝑆𝑋𝑦})
19 ne0i 3879 . . . . . 6 (𝐴 ∈ {𝑦𝑆𝑋𝑦} → {𝑦𝑆𝑋𝑦} ≠ ∅)
2018, 19syl 17 . . . . 5 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ≠ ∅)
21 intex 4741 . . . . 5 ({𝑦𝑆𝑋𝑦} ≠ ∅ ↔ {𝑦𝑆𝑋𝑦} ∈ V)
2220, 21sylib 206 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ∈ V)
23 sseq1 3588 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
2423rabbidv 3163 . . . . . 6 (𝑥 = 𝑋 → {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
2524inteqd 4409 . . . . 5 (𝑥 = 𝑋 {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
26 eqid 2609 . . . . 5 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})
2725, 26fvmptg 6173 . . . 4 ((𝑋 ∈ 𝒫 𝐴 {𝑦𝑆𝑋𝑦} ∈ V) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
2810, 22, 27syl2anc 690 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
299, 28eqtrd 2643 . 2 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
304, 29sylan2br 491 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  Vcvv 3172  wss 3539  c0 3873  𝒫 cpw 4107   cint 4404  cmpt 4637  cfv 5789  Atomscatm 33351  PSubSpcpsubsp 33583  PClcpclN 33974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-psubsp 33590  df-pclN 33975
This theorem is referenced by:  pclclN  33978  elpclN  33979  elpcliN  33980  pclssN  33981  pclssidN  33982  pclidN  33983
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