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Theorem pointsetN 35548
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 3352 . 2 (𝐾𝐵𝐾 ∈ V)
2 pointset.p . . 3 𝑃 = (Points‘𝐾)
3 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 pointset.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2812 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65rexeqdv 3284 . . . . 5 (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎𝐴 𝑝 = {𝑎}))
76abbidv 2879 . . . 4 (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
8 df-pointsN 35309 . . . 4 Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
9 fvex 6363 . . . . . 6 (Atoms‘𝐾) ∈ V
104, 9eqeltri 2835 . . . . 5 𝐴 ∈ V
1110abrexex 7307 . . . 4 {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}} ∈ V
127, 8, 11fvmpt 6445 . . 3 (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
132, 12syl5eq 2806 . 2 (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
141, 13syl 17 1 (𝐾𝐵𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  {csn 4321  cfv 6049  Atomscatm 35071  PointscpointsN 35302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-pointsN 35309
This theorem is referenced by:  ispointN  35549
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