Theorem pointsetN 40548

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	⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		pointset.p	. . 3 ⊢ 𝑃 = (Points‘𝐾)
3		fveq2 6885	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		pointset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2818	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	rexeqdv 3326	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}))
7	6	abbidv 2831	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
8		df-pointsN 40309	. . . 4 ⊢ Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
9	4	fvexi 6899	. . . . 5 ⊢ 𝐴 ∈ V
10	9	abrexex 7961	. . . 4 ⊢ {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}} ∈ V
11	7, 8, 10	fvmpt 6993	. . 3 ⊢ (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
12	2, 11	eqtrid 2812	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
13	1, 12	syl 18	1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {cab 2743  ∃wrex 3091  Vcvv 3457  {csn 4591  ‘cfv 6540  Atomscatm 40070  PointscpointsN 40302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-pointsN 40309

This theorem is used by:  ispointN  40549