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Theorem pointsetN 38254
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 3465 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Pointsβ€˜πΎ)
3 fveq2 6846 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
4 pointset.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2791 . . . . . 6 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
65rexeqdv 3313 . . . . 5 (π‘˜ = 𝐾 β†’ (βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž} ↔ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}))
76abbidv 2802 . . . 4 (π‘˜ = 𝐾 β†’ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}} = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
8 df-pointsN 38015 . . . 4 Points = (π‘˜ ∈ V ↦ {𝑝 ∣ βˆƒπ‘Ž ∈ (Atomsβ€˜π‘˜)𝑝 = {π‘Ž}})
94fvexi 6860 . . . . 5 𝐴 ∈ V
109abrexex 7899 . . . 4 {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}} ∈ V
117, 8, 10fvmpt 6952 . . 3 (𝐾 ∈ V β†’ (Pointsβ€˜πΎ) = {𝑝 ∣ βˆƒπ‘Ž ∈ 𝐴 𝑝 = {π‘Ž}})
122, 11eqtrid 2785 . 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 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070  Vcvv 3447  {csn 4590  β€˜cfv 6500  Atomscatm 37775  PointscpointsN 38008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-pointsN 38015
This theorem is referenced by:  ispointN  38255
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