Theorem pointsetN 37682

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 3440	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		pointset.p	. . 3 ⊢ 𝑃 = (Points‘𝐾)
3		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		pointset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2797	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	rexeqdv 3340	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}))
7	6	abbidv 2808	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
8		df-pointsN 37443	. . . 4 ⊢ Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
9	4	fvexi 6770	. . . . 5 ⊢ 𝐴 ∈ V
10	9	abrexex 7778	. . . 4 ⊢ {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}} ∈ V
11	7, 8, 10	fvmpt 6857	. . 3 ⊢ (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
12	2, 11	syl5eq 2791	. 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 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422  {csn 4558  ‘cfv 6418  Atomscatm 37204  PointscpointsN 37436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-pointsN 37443

This theorem is referenced by:  ispointN  37683