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Theorem pointsetN 30475
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 2956 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 pointset.p . . 3  |-  P  =  ( Points `  K )
3 fveq2 5720 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pointset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65rexeqdv 2903 . . . . 5  |-  ( k  =  K  ->  ( E. a  e.  ( Atoms `  k ) p  =  { a }  <->  E. a  e.  A  p  =  { a } ) )
76abbidv 2549 . . . 4  |-  ( k  =  K  ->  { p  |  E. a  e.  (
Atoms `  k ) p  =  { a } }  =  { p  |  E. a  e.  A  p  =  { a } } )
8 df-pointsN 30236 . . . 4  |-  Points  =  ( k  e.  _V  |->  { p  |  E. a  e.  ( Atoms `  k )
p  =  { a } } )
9 fvex 5734 . . . . . 6  |-  ( Atoms `  K )  e.  _V
104, 9eqeltri 2505 . . . . 5  |-  A  e. 
_V
1110abrexex 5975 . . . 4  |-  { p  |  E. a  e.  A  p  =  { a } }  e.  _V
127, 8, 11fvmpt 5798 . . 3  |-  ( K  e.  _V  ->  ( Points `
 K )  =  { p  |  E. a  e.  A  p  =  { a } }
)
132, 12syl5eq 2479 . 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948   {csn 3806   ` cfv 5446   Atomscatm 29998   PointscpointsN 30229
This theorem is referenced by:  ispointN  30476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-pointsN 30236
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