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Theorem elpmap 30006
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 30005 . . 3  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { x  e.  A  |  x  .<_  X } )
65eleq2d 2433 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( P  e.  ( M `  X )  <-> 
P  e.  { x  e.  A  |  x  .<_  X } ) )
7 breq1 4128 . . 3  |-  ( x  =  P  ->  (
x  .<_  X  <->  P  .<_  X ) )
87elrab 3009 . 2  |-  ( P  e.  { x  e.  A  |  x  .<_  X }  <->  ( P  e.  A  /\  P  .<_  X ) )
96, 8syl6bb 252 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 176    /\ wa 358    = wceq 1647    e. wcel 1715   {crab 2632   class class class wbr 4125   ` cfv 5358   Basecbs 13356   lecple 13423   Atomscatm 29512   pmapcpmap 29745
This theorem is referenced by:  pmapjoin  30100  pmapjat1  30101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-pmap 29752
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