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Theorem dmprop 5098
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 5096 . 2  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
)
41, 2, 3mp2an 426 1  |-  dom  { <. A ,  B >. , 
<. C ,  D >. }  =  { A ,  C }
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148   _Vcvv 2737   {cpr 3592   <.cop 3594   dom cdm 4622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-dm 4632
This theorem is referenced by:  dmtpop  5099  funtp  5264  fpr  5693
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