Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pointsetN Structured version   Visualization version   GIF version

Theorem pointsetN 39724
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 3499 . 2 (𝐾𝐵𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Points‘𝐾)
3 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pointset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2793 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65rexeqdv 3325 . . . . 5 (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎𝐴 𝑝 = {𝑎}))
76abbidv 2806 . . . 4 (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
8 df-pointsN 39485 . . . 4 Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
94fvexi 6921 . . . . 5 𝐴 ∈ V
109abrexex 7986 . . . 4 {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}} ∈ V
117, 8, 10fvmpt 7016 . . 3 (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
122, 11eqtrid 2787 . 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 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  {csn 4631  cfv 6563  Atomscatm 39245  PointscpointsN 39478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-pointsN 39485
This theorem is referenced by:  ispointN  39725
  Copyright terms: Public domain W3C validator