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Theorem op1sta 5488
Description: Extract the first member of an ordered pair. (See op2nda 5491 to extract the second member, op1stb 4717 for an alternate version, and op1st 6789 for the preferred 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 5480 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 4254 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 4260 . 2  |-  U. { A }  =  A
63, 5eqtri 2496 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   <.cop 4033   U.cuni 4245   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-dm 5009
This theorem is referenced by:  elxp4  6725  op1st  6789  fo1st  6801  f1stres  6803  xpassen  7608  xpdom2  7609
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