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Theorem diaeldm 31226
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 31225 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
x  e.  B  |  x  .<_  W } )
65eleq2d 2350 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { x  e.  B  |  x  .<_  W } ) )
7 breq1 4026 . . 3  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
87elrab 2923 . 2  |-  ( X  e.  { x  e.  B  |  x  .<_  W }  <->  ( X  e.  B  /\  X  .<_  W ) )
96, 8syl6bb 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   dom cdm 4689   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   DIsoAcdia 31218
This theorem is referenced by:  diadmclN  31227  diadmleN  31228  dia0eldmN  31230  dia1eldmN  31231  diaf11N  31239  diaglbN  31245  diaintclN  31248  diasslssN  31249  docaclN  31314  doca2N  31316  djajN  31327  dibval2  31334  dibeldmN  31348
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-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-pr 4214
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-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-disoa 31219
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