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Theorem dmprop 5488
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 5486 . 2  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
)
41, 2, 3mp2an 672 1  |-  dom  { <. A ,  B >. , 
<. C ,  D >. }  =  { A ,  C }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3118   {cpr 4034   <.cop 4038   dom cdm 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4453  df-dm 5014
This theorem is referenced by:  dmtpop  5489  funtp  5645  fpr  6079  fnprb  6129  fnprOLD  6130  hashfun  12471  ex-dm  24952
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