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Theorem dmprop 5425
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 5423 . 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 1370    e. wcel 1758   _Vcvv 3078   {cpr 3990   <.cop 3994   dom cdm 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-dm 4961
This theorem is referenced by:  dmtpop  5426  funtp  5581  fpr  6002  fnprb  6048  fnprOLD  6049  hashfun  12321  ex-dm  23825
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