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Theorem pointsetN 39068
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 3485 . 2 (𝐾𝐵𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Points‘𝐾)
3 fveq2 6881 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pointset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2782 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65rexeqdv 3318 . . . . 5 (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎𝐴 𝑝 = {𝑎}))
76abbidv 2793 . . . 4 (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
8 df-pointsN 38829 . . . 4 Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
94fvexi 6895 . . . . 5 𝐴 ∈ V
109abrexex 7942 . . . 4 {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}} ∈ V
117, 8, 10fvmpt 6988 . . 3 (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
122, 11eqtrid 2776 . 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 1533  wcel 2098  {cab 2701  wrex 3062  Vcvv 3466  {csn 4620  cfv 6533  Atomscatm 38589  PointscpointsN 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-pointsN 38829
This theorem is referenced by:  ispointN  39069
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