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Theorem elpmap 30252
Description: Member of a projective map. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapfval.b  |-  B  =  ( Base `  K
)
pmapfval.l  |-  .<_  =  ( le `  K )
pmapfval.a  |-  A  =  ( Atoms `  K )
pmapfval.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
elpmap  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( P  e.  ( M `  X )  <-> 
( P  e.  A  /\  P  .<_  X ) ) )

Proof of Theorem elpmap
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4  |-  B  =  ( Base `  K
)
2 pmapfval.l . . . 4  |-  .<_  =  ( le `  K )
3 pmapfval.a . . . 4  |-  A  =  ( Atoms `  K )
4 pmapfval.m . . . 4  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 30251 . . 3  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { x  e.  A  |  x  .<_  X } )
65eleq2d 2479 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( P  e.  ( M `  X )  <-> 
P  e.  { x  e.  A  |  x  .<_  X } ) )
7 breq1 4183 . . 3  |-  ( x  =  P  ->  (
x  .<_  X  <->  P  .<_  X ) )
87elrab 3060 . 2  |-  ( P  e.  { x  e.  A  |  x  .<_  X }  <->  ( P  e.  A  /\  P  .<_  X ) )
96, 8syl6bb 253 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( P  e.  ( M `  X )  <-> 
( P  e.  A  /\  P  .<_  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2678   class class class wbr 4180   ` cfv 5421   Basecbs 13432   lecple 13499   Atomscatm 29758   pmapcpmap 29991
This theorem is referenced by:  pmapjoin  30346  pmapjat1  30347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-pmap 29998
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