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Theorem op1sta 5131
Description: Extract the first member of an ordered pair. (See op2nda 5134 to extract the second member and op1stb 4499 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
Hypotheses
Ref Expression
cnvsn.1  |-  A  e. 
_V
cnvsn.2  |-  B  e. 
_V
Assertion
Ref Expression
op1sta  |-  U. dom  {
<. A ,  B >. }  =  A

Proof of Theorem op1sta
StepHypRef Expression
1 cnvsn.2 . . . 4  |-  B  e. 
_V
21dmsnop 5123 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 3837 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 3843 . 2  |-  U. { A }  =  A
63, 5eqtri 2210 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3610   <.cop 3613   U.cuni 3827   dom cdm 4647
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-dm 4657
This theorem is referenced by:  op1st  6175  fo1st  6186  f1stres  6188  xpassen  6860  xpdom2  6861
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