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Mirrors > Home > MPE Home > Th. List > Mathboxes > pointsetN | Structured version Visualization version GIF version |
Description: The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pointset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pointset.p | ⊢ 𝑃 = (Points‘𝐾) |
Ref | Expression |
---|---|
pointsetN | ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3512 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
2 | pointset.p | . . 3 ⊢ 𝑃 = (Points‘𝐾) | |
3 | fveq2 6669 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
4 | pointset.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
6 | 5 | rexeqdv 3416 | . . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎} ↔ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎})) |
7 | 6 | abbidv 2885 | . . . 4 ⊢ (𝑘 = 𝐾 → {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}} = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}}) |
8 | df-pointsN 36637 | . . . 4 ⊢ Points = (𝑘 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ (Atoms‘𝑘)𝑝 = {𝑎}}) | |
9 | 4 | fvexi 6683 | . . . . 5 ⊢ 𝐴 ∈ V |
10 | 9 | abrexex 7662 | . . . 4 ⊢ {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}} ∈ V |
11 | 7, 8, 10 | fvmpt 6767 | . . 3 ⊢ (𝐾 ∈ V → (Points‘𝐾) = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}}) |
12 | 2, 11 | syl5eq 2868 | . 2 ⊢ (𝐾 ∈ V → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}}) |
13 | 1, 12 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 Vcvv 3494 {csn 4566 ‘cfv 6354 Atomscatm 36398 PointscpointsN 36630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-pointsN 36637 |
This theorem is referenced by: ispointN 36877 |
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