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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 3465 | . 2 β’ (πΎ β π΅ β πΎ β V) | |
2 | pointset.p | . . 3 β’ π = (PointsβπΎ) | |
3 | fveq2 6846 | . . . . . . 7 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | pointset.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2791 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | 5 | rexeqdv 3313 | . . . . 5 β’ (π = πΎ β (βπ β (Atomsβπ)π = {π} β βπ β π΄ π = {π})) |
7 | 6 | abbidv 2802 | . . . 4 β’ (π = πΎ β {π β£ βπ β (Atomsβπ)π = {π}} = {π β£ βπ β π΄ π = {π}}) |
8 | df-pointsN 38015 | . . . 4 β’ Points = (π β V β¦ {π β£ βπ β (Atomsβπ)π = {π}}) | |
9 | 4 | fvexi 6860 | . . . . 5 β’ π΄ β V |
10 | 9 | abrexex 7899 | . . . 4 β’ {π β£ βπ β π΄ π = {π}} β V |
11 | 7, 8, 10 | fvmpt 6952 | . . 3 β’ (πΎ β V β (PointsβπΎ) = {π β£ βπ β π΄ π = {π}}) |
12 | 2, 11 | eqtrid 2785 | . 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 1542 β wcel 2107 {cab 2710 βwrex 3070 Vcvv 3447 {csn 4590 βcfv 6500 Atomscatm 37775 PointscpointsN 38008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-pointsN 38015 |
This theorem is referenced by: ispointN 38255 |
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