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Theorem pointsetN 37053
 Description: The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
pointset.a 𝐴 = (Atoms‘𝐾)
pointset.p 𝑃 = (Points‘𝐾)
Assertion
Ref Expression
pointsetN (𝐾𝐵𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
Distinct variable groups:   𝑝,𝑎,𝐴   𝐾,𝑝
Allowed substitution hints:   𝐵(𝑝,𝑎)   𝑃(𝑝,𝑎)   𝐾(𝑎)

Proof of Theorem pointsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3459 . 2 (𝐾𝐵𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Points‘𝐾)
3 fveq2 6645 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pointset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2851 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65rexeqdv 3365 . . . . 5 (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎𝐴 𝑝 = {𝑎}))
76abbidv 2862 . . . 4 (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
8 df-pointsN 36814 . . . 4 Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
94fvexi 6659 . . . . 5 𝐴 ∈ V
109abrexex 7647 . . . 4 {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}} ∈ V
117, 8, 10fvmpt 6745 . . 3 (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
122, 11syl5eq 2845 . 2 (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
131, 12syl 17 1 (𝐾𝐵𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441  {csn 4525  ‘cfv 6324  Atomscatm 36575  PointscpointsN 36807 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 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 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 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-pointsN 36814 This theorem is referenced by:  ispointN  37054
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