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Theorem pmapssat 30557
Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
Hypotheses
Ref Expression
pmapssat.b  |-  B  =  ( Base `  K
)
pmapssat.a  |-  A  =  ( Atoms `  K )
pmapssat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapssat  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )

Proof of Theorem pmapssat
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 pmapssat.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2437 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 pmapssat.a . . 3  |-  A  =  ( Atoms `  K )
4 pmapssat.m . . 3  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 30555 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  A  |  p
( le `  K
) X } )
6 ssrab2 3429 . 2  |-  { p  e.  A  |  p
( le `  K
) X }  C_  A
75, 6syl6eqss 3399 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2710    C_ wss 3321   class class class wbr 4213   ` cfv 5455   Basecbs 13470   lecple 13537   Atomscatm 30062   pmapcpmap 30295
This theorem is referenced by:  pmapssbaN  30558  pmapglb2N  30569  pmapglb2xN  30570  pmapjoin  30650  pmapjat1  30651  pmapjat2  30652  pmapjlln1  30653  hlmod1i  30654  polpmapN  30710  2pmaplubN  30724  pmapj2N  30727  pmapocjN  30728  polatN  30729  pmapsubclN  30744  ispsubcl2N  30745  pl42lem2N  30778  pl42lem3N  30779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-pmap 30302
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