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Theorem elpmapat 29953
Description: Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapat.a  |-  A  =  ( Atoms `  K )
pmapat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
elpmapat  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( X  e.  ( M `  P )  <-> 
X  =  P ) )

Proof of Theorem elpmapat
StepHypRef Expression
1 pmapat.a . . . 4  |-  A  =  ( Atoms `  K )
2 pmapat.m . . . 4  |-  M  =  ( pmap `  K
)
31, 2pmapat 29952 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { P } )
43eleq2d 2350 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( X  e.  ( M `  P )  <-> 
X  e.  { P } ) )
5 elsnc2g 3668 . . 3  |-  ( P  e.  A  ->  ( X  e.  { P } 
<->  X  =  P ) )
65adantl 452 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( X  e.  { P }  <->  X  =  P
) )
74, 6bitrd 244 1  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( X  e.  ( M `  P )  <-> 
X  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   ` cfv 5255   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686
This theorem is referenced by:  pmapjat1  30042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693
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