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Theorem pointsetN 33306
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 3054 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 pointset.p . . 3  |-  P  =  ( Points `  K )
3 fveq2 5865 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pointset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2503 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65rexeqdv 2994 . . . . 5  |-  ( k  =  K  ->  ( E. a  e.  ( Atoms `  k ) p  =  { a }  <->  E. a  e.  A  p  =  { a } ) )
76abbidv 2569 . . . 4  |-  ( k  =  K  ->  { p  |  E. a  e.  (
Atoms `  k ) p  =  { a } }  =  { p  |  E. a  e.  A  p  =  { a } } )
8 df-pointsN 33067 . . . 4  |-  Points  =  ( k  e.  _V  |->  { p  |  E. a  e.  ( Atoms `  k )
p  =  { a } } )
9 fvex 5875 . . . . . 6  |-  ( Atoms `  K )  e.  _V
104, 9eqeltri 2525 . . . . 5  |-  A  e. 
_V
1110abrexex 6767 . . . 4  |-  { p  |  E. a  e.  A  p  =  { a } }  e.  _V
127, 8, 11fvmpt 5948 . . 3  |-  ( K  e.  _V  ->  ( Points `
 K )  =  { p  |  E. a  e.  A  p  =  { a } }
)
132, 12syl5eq 2497 . 2  |-  ( K  e.  _V  ->  P  =  { p  |  E. a  e.  A  p  =  { a } }
)
141, 13syl 17 1  |-  ( K  e.  B  ->  P  =  { p  |  E. a  e.  A  p  =  { a } }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045   {csn 3968   ` cfv 5582   Atomscatm 32829   PointscpointsN 33060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-pointsN 33067
This theorem is referenced by:  ispointN  33307
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