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	⊢ (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 3352	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ 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)
5	3, 4	syl6eqr 2812	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	rexeqdv 3284	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}))
7	6	abbidv 2879	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
8		df-pointsN 35309	. . . 4 ⊢ Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}})
9		fvex 6363	. . . . . 6 ⊢ (Atoms‘𝐾) ∈ V
10	4, 9	eqeltri 2835	. . . . 5 ⊢ 𝐴 ∈ V
11	10	abrexex 7307	. . . 4 ⊢ {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}} ∈ V
12	7, 8, 11	fvmpt 6445	. . 3 ⊢ (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
13	2, 12	syl5eq 2806	. 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
14	1, 13	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})

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