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Theorem op1sta 4988
Description: Extract the first member of an ordered pair. (See op2nda 4991 to extract the second member and op1stb 4367 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 4980 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 3714 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 3720 . 2  |-  U. { A }  =  A
63, 5eqtri 2136 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   _Vcvv 2658   {csn 3495   <.cop 3498   U.cuni 3704   dom cdm 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-dm 4517
This theorem is referenced by:  op1st  6010  fo1st  6021  f1stres  6023  xpassen  6690  xpdom2  6691
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