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Theorem dmprop 5302
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 5300 . 2  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
)
41, 2, 3mp2an 665 1  |-  dom  { <. A ,  B >. , 
<. C ,  D >. }  =  { A ,  C }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   _Vcvv 2962   {cpr 3867   <.cop 3871   dom cdm 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-dm 4837
This theorem is referenced by:  dmtpop  5303  funtp  5458  fpr  5877  fnprb  5923  fnprOLD  5924  hashfun  12183  ex-dm  23469
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