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Theorem op1sta 5496
Description: Extract the first member of an ordered pair. (See op2nda 5499 to extract the second member, op1stb 4726 for an alternate version, and op1st 6807 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 5488 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 4260 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 4266 . 2  |-  U. { A }  =  A
63, 5eqtri 2486 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038   U.cuni 4251   dom cdm 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018
This theorem is referenced by:  elxp4  6743  op1st  6807  fo1st  6819  f1stres  6821  xpassen  7630  xpdom2  7631
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