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Theorem op1sta 4832
Description: Extract the first member of an ordered pair. (See op2nda 4835 to extract the second member and op1stb 4235 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 4824 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 3619 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 3625 . 2  |-  U. { A }  =  A
63, 5eqtri 2102 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   _Vcvv 2602   {csn 3406   <.cop 3409   U.cuni 3609   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-dm 4381
This theorem is referenced by:  op1st  5804  fo1st  5815  f1stres  5817  xpassen  6374  xpdom2  6375
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