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Theorem dmprop 5347
Description: The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
dmsnop.1  |-  B  e. 
_V
dmprop.1  |-  D  e. 
_V
Assertion
Ref Expression
dmprop  |-  dom  { <. A ,  B >. , 
<. C ,  D >. }  =  { A ,  C }

Proof of Theorem dmprop
StepHypRef Expression
1 dmsnop.1 . 2  |-  B  e. 
_V
2 dmprop.1 . 2  |-  D  e. 
_V
3 dmpropg 5345 . 2  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
)
41, 2, 3mp2an 655 1  |-  dom  { <. A ,  B >. , 
<. C ,  D >. }  =  { A ,  C }
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   {cpr 3817   <.cop 3819   dom cdm 4880
This theorem is referenced by:  dmtpop  5348  funtp  5505  fpr  5916  fnpr  5952  fnprOLD  5953  hashfun  11702  ex-dm  21749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-dm 4890
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