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Theorem pointsetN 39214
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 = (Atomsβ€˜πΎ)
pointset.p 𝑃 = (Pointsβ€˜πΎ)
Assertion
Ref Expression
pointsetN (𝐾 ∈ 𝐡 β†’ 𝑃 = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
Distinct variable groups:   𝑝,π‘Ž,𝐴   𝐾,𝑝
Allowed substitution hints:   𝐡(𝑝,π‘Ž)   𝑃(𝑝,π‘Ž)   𝐾(π‘Ž)

Proof of Theorem pointsetN
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3490 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Pointsβ€˜πΎ)
3 fveq2 6897 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 pointset.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2786 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65rexeqdv 3323 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž} ↔ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}))
76abbidv 2797 . . . 4 (π‘˜ = 𝐾 β†’ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}} = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
8 df-pointsN 38975 . . . 4 Points = (π‘˜ ∈ V ↦ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}})
94fvexi 6911 . . . . 5 𝐴 ∈ V
109abrexex 7966 . . . 4 {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}} ∈ V
117, 8, 10fvmpt 7005 . . 3 (𝐾 ∈ V β†’ (Pointsβ€˜πΎ) = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
122, 11eqtrid 2780 . 2 (𝐾 ∈ V β†’ 𝑃 = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
131, 12syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝑃 = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  {cab 2705  βˆƒwrex 3067  Vcvv 3471  {csn 4629  β€˜cfv 6548  Atomscatm 38735  PointscpointsN 38968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-pointsN 38975
This theorem is referenced by:  ispointN  39215
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