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Theorem diaeldm 31772
Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
Hypotheses
Ref Expression
diafn.b  |-  B  =  ( Base `  K
)
diafn.l  |-  .<_  =  ( le `  K )
diafn.h  |-  H  =  ( LHyp `  K
)
diafn.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaeldm  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X  .<_  W ) ) )

Proof of Theorem diaeldm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 diafn.b . . . 4  |-  B  =  ( Base `  K
)
2 diafn.l . . . 4  |-  .<_  =  ( le `  K )
3 diafn.h . . . 4  |-  H  =  ( LHyp `  K
)
4 diafn.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diadm 31771 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
x  e.  B  |  x  .<_  W } )
65eleq2d 2503 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { x  e.  B  |  x  .<_  W } ) )
7 breq1 4208 . . 3  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
87elrab 3085 . 2  |-  ( X  e.  { x  e.  B  |  x  .<_  W }  <->  ( X  e.  B  /\  X  .<_  W ) )
96, 8syl6bb 253 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X  .<_  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2702   class class class wbr 4205   dom cdm 4871   ` cfv 5447   Basecbs 13462   lecple 13529   LHypclh 30719   DIsoAcdia 31764
This theorem is referenced by:  diadmclN  31773  diadmleN  31774  dia0eldmN  31776  dia1eldmN  31777  diaf11N  31785  diaglbN  31791  diaintclN  31794  diasslssN  31795  docaclN  31860  doca2N  31862  djajN  31873  dibval2  31880  dibeldmN  31894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-disoa 31765
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