Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pointsetN Structured version   Unicode version

Theorem pointsetN 32739
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 3067 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 pointset.p . . 3  |-  P  =  ( Points `  K )
3 fveq2 5805 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 pointset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2461 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65rexeqdv 3010 . . . . 5  |-  ( k  =  K  ->  ( E. a  e.  ( Atoms `  k ) p  =  { a }  <->  E. a  e.  A  p  =  { a } ) )
76abbidv 2538 . . . 4  |-  ( k  =  K  ->  { p  |  E. a  e.  (
Atoms `  k ) p  =  { a } }  =  { p  |  E. a  e.  A  p  =  { a } } )
8 df-pointsN 32500 . . . 4  |-  Points  =  ( k  e.  _V  |->  { p  |  E. a  e.  ( Atoms `  k )
p  =  { a } } )
9 fvex 5815 . . . . . 6  |-  ( Atoms `  K )  e.  _V
104, 9eqeltri 2486 . . . . 5  |-  A  e. 
_V
1110abrexex 6712 . . . 4  |-  { p  |  E. a  e.  A  p  =  { a } }  e.  _V
127, 8, 11fvmpt 5888 . . 3  |-  ( K  e.  _V  ->  ( Points `
 K )  =  { p  |  E. a  e.  A  p  =  { a } }
)
132, 12syl5eq 2455 . 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 1405    e. wcel 1842   {cab 2387   E.wrex 2754   _Vcvv 3058   {csn 3971   ` cfv 5525   Atomscatm 32262   PointscpointsN 32493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-pointsN 32500
This theorem is referenced by:  ispointN  32740
  Copyright terms: Public domain W3C validator