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 37492
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 3426 . 2 (𝐾𝐵𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Points‘𝐾)
3 fveq2 6717 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pointset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2796 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65rexeqdv 3326 . . . . 5 (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎𝐴 𝑝 = {𝑎}))
76abbidv 2807 . . . 4 (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
8 df-pointsN 37253 . . . 4 Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
94fvexi 6731 . . . . 5 𝐴 ∈ V
109abrexex 7735 . . . 4 {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}} ∈ V
117, 8, 10fvmpt 6818 . . 3 (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
122, 11syl5eq 2790 . 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 1543  wcel 2110  {cab 2714  wrex 3062  Vcvv 3408  {csn 4541  cfv 6380  Atomscatm 37014  PointscpointsN 37246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-pointsN 37253
This theorem is referenced by:  ispointN  37493
  Copyright terms: Public domain W3C validator