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Mathbox for Norm Megill |
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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 | β’ π΄ = (AtomsβπΎ) |
pointset.p | β’ π = (PointsβπΎ) |
Ref | Expression |
---|---|
pointsetN | β’ (πΎ β π΅ β π = {π β£ βπ β π΄ π = {π}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3485 | . 2 β’ (πΎ β π΅ β πΎ β V) | |
2 | pointset.p | . . 3 β’ π = (PointsβπΎ) | |
3 | fveq2 6882 | . . . . . . 7 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | pointset.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2782 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | 5 | rexeqdv 3318 | . . . . 5 β’ (π = πΎ β (βπ β (Atomsβπ)π = {π} β βπ β π΄ π = {π})) |
7 | 6 | abbidv 2793 | . . . 4 β’ (π = πΎ β {π β£ βπ β (Atomsβπ)π = {π}} = {π β£ βπ β π΄ π = {π}}) |
8 | df-pointsN 38877 | . . . 4 β’ Points = (π β V β¦ {π β£ βπ β (Atomsβπ)π = {π}}) | |
9 | 4 | fvexi 6896 | . . . . 5 β’ π΄ β V |
10 | 9 | abrexex 7943 | . . . 4 β’ {π β£ βπ β π΄ π = {π}} β V |
11 | 7, 8, 10 | fvmpt 6989 | . . 3 β’ (πΎ β V β (PointsβπΎ) = {π β£ βπ β π΄ π = {π}}) |
12 | 2, 11 | eqtrid 2776 | . 2 β’ (πΎ β V β π = {π β£ βπ β π΄ π = {π}}) |
13 | 1, 12 | syl 17 | 1 β’ (πΎ β π΅ β π = {π β£ βπ β π΄ π = {π}}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {cab 2701 βwrex 3062 Vcvv 3466 {csn 4621 βcfv 6534 Atomscatm 38637 PointscpointsN 38870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-pointsN 38877 |
This theorem is referenced by: ispointN 39117 |
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