Theorem pointsetN 35027

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 e. B -> P = { p | E. a e. 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 3212	. 2 |- ( K e. B -> K e. _V )
2		pointset.p	. . 3 |- P = ( Points ` K )
3		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		pointset.a	. . . . . . 7 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = A )
6	5	rexeqdv 3145	. . . . 5 |- ( k = K -> ( E. a e. ( Atoms ` k ) p = { a } <-> E. a e. A p = { a } ) )
7	6	abbidv 2741	. . . 4 |- ( k = K -> { p | E. a e. ( Atoms ` k ) p = { a } } = { p | E. a e. A p = { a } } )
8		df-pointsN 34788	. . . 4 |- Points = ( k e. _V |-> { p | E. a e. ( Atoms ` k ) p = { a } } )
9		fvex 6201	. . . . . 6 |- ( Atoms ` K ) e. _V
10	4, 9	eqeltri 2697	. . . . 5 |- A e. _V
11	10	abrexex 7141	. . . 4 |- { p | E. a e. A p = { a } } e. _V
12	7, 8, 11	fvmpt 6282	. . 3 |- ( K e. _V -> ( Points ` K ) = { p | E. a e. A p = { a } } )
13	2, 12	syl5eq 2668	. 2 |- ( K e. _V -> P = { p | E. a e. A p = { a } } )
14	1, 13	syl 17	1 |- ( K e. B -> P = { p | E. a e. A p = { a } } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   {csn 4177   ` cfv 5888   Atomscatm 34550   PointscpointsN 34781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pointsN 34788

This theorem is referenced by:  ispointN  35028