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Theorem pointsetN 34412
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.
StepHypRef Expression
1 elex 3115 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 pointset.p . . 3  |-  P  =  ( Points `  K )
3 fveq2 5857 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pointset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2519 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65rexeqdv 3058 . . . . 5  |-  ( k  =  K  ->  ( E. a  e.  ( Atoms `  k ) p  =  { a }  <->  E. a  e.  A  p  =  { a } ) )
76abbidv 2596 . . . 4  |-  ( k  =  K  ->  { p  |  E. a  e.  (
Atoms `  k ) p  =  { a } }  =  { p  |  E. a  e.  A  p  =  { a } } )
8 df-pointsN 34173 . . . 4  |-  Points  =  ( k  e.  _V  |->  { p  |  E. a  e.  ( Atoms `  k )
p  =  { a } } )
9 fvex 5867 . . . . . 6  |-  ( Atoms `  K )  e.  _V
104, 9eqeltri 2544 . . . . 5  |-  A  e. 
_V
1110abrexex 6748 . . . 4  |-  { p  |  E. a  e.  A  p  =  { a } }  e.  _V
127, 8, 11fvmpt 5941 . . 3  |-  ( K  e.  _V  ->  ( Points `
 K )  =  { p  |  E. a  e.  A  p  =  { a } }
)
132, 12syl5eq 2513 . 2  |-  ( K  e.  _V  ->  P  =  { p  |  E. a  e.  A  p  =  { a } }
)
141, 13syl 16 1  |-  ( K  e.  B  ->  P  =  { p  |  E. a  e.  A  p  =  { a } }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   {cab 2445   E.wrex 2808   _Vcvv 3106   {csn 4020   ` cfv 5579   Atomscatm 33935   PointscpointsN 34166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-pointsN 34173
This theorem is referenced by:  ispointN  34413
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