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Theorem pmapssat 34573
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 2467 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 pmapssat.a . . 3  |-  A  =  ( Atoms `  K )
4 pmapssat.m . . 3  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 34571 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  A  |  p
( le `  K
) X } )
6 ssrab2 3585 . 2  |-  { p  e.  A  |  p
( le `  K
) X }  C_  A
75, 6syl6eqss 3554 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   class class class wbr 4447   ` cfv 5588   Basecbs 14490   lecple 14562   Atomscatm 34078   pmapcpmap 34311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-pmap 34318
This theorem is referenced by:  pmapssbaN  34574  pmapglb2N  34585  pmapglb2xN  34586  pmapjoin  34666  pmapjat1  34667  pmapjat2  34668  pmapjlln1  34669  hlmod1i  34670  polpmapN  34726  2pmaplubN  34740  pmapj2N  34743  pmapocjN  34744  polatN  34745  pmapsubclN  34760  ispsubcl2N  34761  pl42lem2N  34794  pl42lem3N  34795
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