Theorem pointset 17222

Description: The set of points in a Hilbert lattice.

Hypotheses

Ref	Expression
pointset.a	|- A = (AtomsNEW` K)
pointset.p	|- P = (Points` K)

Assertion

Ref	Expression
pointset	|- (K e. B -> P = {p | E.a e. A p = {a}})

Distinct variable groups:   p,a,A   K,p

Proof of Theorem pointset

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. B -> K e. _V)
2		fveq2 4681	. . . . . . 7 |- (h = K -> (AtomsNEW` h) = (AtomsNEW` K))
3		pointset.a	. . . . . . 7 |- A = (AtomsNEW` K)
4	2, 3	syl6eqr 1946	. . . . . 6 |- (h = K -> (AtomsNEW` h) = A)
5	4	rexeqdv 2270	. . . . 5 |- (h = K -> (E.a e. (AtomsNEW` h)p = {a} <-> E.a e. A p = {a}))
6	5	abbidv 2008	. . . 4 |- (h = K -> {p | E.a e. (AtomsNEW` h)p = {a}} = {p | E.a e. A p = {a}})
7		df-points 17216	. . . 4 |- Points = (h e. _V |-> {p | E.a e. (AtomsNEW` h)p = {a}})
8		fvex 4689	. . . . . 6 |- (AtomsNEW` K) e. _V
9	3, 8	eqeltri 1967	. . . . 5 |- A e. _V
10	9	abrexex 4836	. . . 4 |- {p | E.a e. A p = {a}} e. _V
11	6, 7, 10	fvmpt 5015	. . 3 |- (K e. _V -> (Points` K) = {p | E.a e. A p = {a}})
12		pointset.p	. . 3 |- P = (Points` K)
13	11, 12	syl5eq 1940	. 2 |- (K e. _V -> P = {p | E.a e. A p = {a}})
14	1, 13	syl 12	1 |- (K e. B -> P = {p | E.a e. A p = {a}})

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292  {csn 3044  ` cfv 3998  AtomsNEWcatm 16981  Pointscpoints 17212

This theorem is referenced by:  ispoint 17223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-mpt 5006  df-points 17216

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