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 = (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 3490	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		pointset.p	. . 3 ⊢ 𝑃 = (Points‘𝐾)
3		fveq2 6897	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		pointset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2786	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	rexeqdv 3323	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}))
7	6	abbidv 2797	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
8		df-pointsN 38975	. . . 4 ⊢ Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
9	4	fvexi 6911	. . . . 5 ⊢ 𝐴 ∈ V
10	9	abrexex 7966	. . . 4 ⊢ {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}} ∈ V
11	7, 8, 10	fvmpt 7005	. . 3 ⊢ (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
12	2, 11	eqtrid 2780	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
13	1, 12	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {cab 2705  ∃wrex 3067  Vcvv 3471  {csn 4629  ‘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