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Theorem pointsetN 29906
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 2900 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 pointset.p . . 3  |-  P  =  ( Points `  K )
3 fveq2 5661 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pointset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2430 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65rexeqdv 2847 . . . . 5  |-  ( k  =  K  ->  ( E. a  e.  ( Atoms `  k ) p  =  { a }  <->  E. a  e.  A  p  =  { a } ) )
76abbidv 2494 . . . 4  |-  ( k  =  K  ->  { p  |  E. a  e.  (
Atoms `  k ) p  =  { a } }  =  { p  |  E. a  e.  A  p  =  { a } } )
8 df-pointsN 29667 . . . 4  |-  Points  =  ( k  e.  _V  |->  { p  |  E. a  e.  ( Atoms `  k )
p  =  { a } } )
9 fvex 5675 . . . . . 6  |-  ( Atoms `  K )  e.  _V
104, 9eqeltri 2450 . . . . 5  |-  A  e. 
_V
1110abrexex 5915 . . . 4  |-  { p  |  E. a  e.  A  p  =  { a } }  e.  _V
127, 8, 11fvmpt 5738 . . 3  |-  ( K  e.  _V  ->  ( Points `
 K )  =  { p  |  E. a  e.  A  p  =  { a } }
)
132, 12syl5eq 2424 . 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 set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {cab 2366   E.wrex 2643   _Vcvv 2892   {csn 3750   ` cfv 5387   Atomscatm 29429   PointscpointsN 29660
This theorem is referenced by:  ispointN  29907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-pointsN 29667
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