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Theorem pmapssat 30570
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 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 pmapssat.a . . 3  |-  A  =  ( Atoms `  K )
4 pmapssat.m . . 3  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 30568 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  A  |  p
( le `  K
) X } )
6 ssrab2 3271 . . 3  |-  { p  e.  A  |  p
( le `  K
) X }  C_  A
76a1i 10 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  { p  e.  A  |  p ( le `  K ) X }  C_  A )
85, 7eqsstrd 3225 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   pmapcpmap 30308
This theorem is referenced by:  pmapssbaN  30571  pmapglb2N  30582  pmapglb2xN  30583  pmapjoin  30663  pmapjat1  30664  pmapjat2  30665  pmapjlln1  30666  hlmod1i  30667  polpmapN  30723  2pmaplubN  30737  pmapj2N  30740  pmapocjN  30741  polatN  30742  pmapsubclN  30757  ispsubcl2N  30758  pl42lem2N  30791  pl42lem3N  30792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pmap 30315
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