Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diaeldm Unicode version

Theorem diaeldm 31848
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 31847 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
x  e.  B  |  x  .<_  W } )
65eleq2d 2363 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { x  e.  B  |  x  .<_  W } ) )
7 breq1 4042 . . 3  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
87elrab 2936 . 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 1632    e. wcel 1696   {crab 2560   class class class wbr 4039   dom cdm 4705   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   DIsoAcdia 31840
This theorem is referenced by:  diadmclN  31849  diadmleN  31850  dia0eldmN  31852  dia1eldmN  31853  diaf11N  31861  diaglbN  31867  diaintclN  31870  diasslssN  31871  docaclN  31936  doca2N  31938  djajN  31949  dibval2  31956  dibeldmN  31970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841
  Copyright terms: Public domain W3C validator