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Theorem pointsetN 39116
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 3485 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Pointsβ€˜πΎ)
3 fveq2 6882 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 pointset.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2782 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65rexeqdv 3318 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž} ↔ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}))
76abbidv 2793 . . . 4 (π‘˜ = 𝐾 β†’ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}} = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
8 df-pointsN 38877 . . . 4 Points = (π‘˜ ∈ V ↦ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}})
94fvexi 6896 . . . . 5 𝐴 ∈ V
109abrexex 7943 . . . 4 {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}} ∈ V
117, 8, 10fvmpt 6989 . . 3 (𝐾 ∈ V β†’ (Pointsβ€˜πΎ) = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
122, 11eqtrid 2776 . 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 1533   ∈ wcel 2098  {cab 2701  βˆƒwrex 3062  Vcvv 3466  {csn 4621  β€˜cfv 6534  Atomscatm 38637  PointscpointsN 38870
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 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-pointsN 38877
This theorem is referenced by:  ispointN  39117
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