Theorem op1sta 3448

Description: Extract the first member of an ordered pair. (See op2nda 3452 to extract the second member, op1stb 2913 for an alternate version, and op1st 4085 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 3328	. . 3 |- dom {<.A, B>.} = {A}
2	1	unieqi 2511	. 2 |- U.dom {<.A, B>.} = U.{A}
3		op1sta.1	. . 3 |- A e. V
4	3	unisn 2517	. 2 |- U.{A} = A
5	2, 4	eqtr 1495	1 |- U.dom {<.A, B>.} = A

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  <.cop 2411  U.cuni 2503  dom cdm 3170

This theorem is referenced by:  op2nda 3452  elxp4 3453  op1st 4085  fo1st 4091  f1stres 4093  xpassen 4441  xpdom2 4442  xpmapenlem2 4497  xpmapenlem4 4499  xpmapenlem5 4500

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-dm 3188

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