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Theorem op1sta 5170
Description: Extract the first member of an ordered pair. (See op2nda 5173 to extract the second member, op1stb 4585 for an alternate version, and op1st 6144 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 5163 . . 3  |-  dom  { <. A ,  B >. }  =  { A }
32unieqi 3853 . 2  |-  U. dom  {
<. A ,  B >. }  =  U. { A }
4 cnvsn.1 . . 3  |-  A  e. 
_V
54unisn 3859 . 2  |-  U. { A }  =  A
63, 5eqtri 2316 1  |-  U. dom  {
<. A ,  B >. }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   U.cuni 3843   dom cdm 4705
This theorem is referenced by:  elxp4  5176  op1st  6144  fo1st  6155  f1stres  6157  xpassen  6972  xpdom2  6973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-dm 4715
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