Theorem op1sta 3434

Description: Extract the first member of an ordered pair. (See op2nda 3438 to extract the second member, op1stb 2903 for an alternate version, and op1st 4069 for the preferred version..) (Contributed by Raph Levien, 4-Dec-2003.)

Hypothesis

Ref	Expression
op1sta.1	|- A e. V

Assertion

Ref	Expression
op1sta	|- U.dom {<.A, B>.} = A

Proof of Theorem op1sta

Step	Hyp	Ref	Expression
1		dmsnop 3317	. . 3 |- dom {<.A, B>.} = {A}
2	1	unieqi 2501	. 2 |- U.dom {<.A, B>.} = U.{A}
3		op1sta.1	. . 3 |- A e. V
4	3	unisn 2507	. 2 |- U.{A} = A
5	2, 4	eqtr 1487	1 |- U.dom {<.A, B>.} = A

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802  {csn 2399  <.cop 2401  U.cuni 2493  dom cdm 3160

This theorem is referenced by:  op2nda 3438  elxp4 3439  op1st 4069  fo1st 4075  f1stres 4077  xpassen 4421  xpdom2 4422  xpmapenlem2 4477  xpmapenlem4 4479  xpmapenlem5 4480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-dm 3178

Copyright terms: Public domain