Theorem op1sta 2635

Description: Extract the first member of an ordered pair. (See op2nda 2639 to extract the second member and op1stb 1992 for an alternate version.) (Contributed by Raph Levien, 4-Dec-03.)

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 2547	. . 3 |- dom {<.A, B>.} = {A}
2	1	unieqi 1928	. 2 |- U.dom {<.A, B>.} = U.{A}
3		op1sta.1	. . 3 |- A e. V
4	3	unisn 1932	. 2 |- U.{A} = A
5	2, 4	eqtr 1119	1 |- U.dom {<.A, B>.} = A

Syntax hints:   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  U.cuni 1919  dom cdm 2410

This theorem is referenced by:  op2nda 2639  elxp4 2640  op1st 3091  fo1st 3094  f1stres 3096  xpassen 3344  xpdom2 3345  xpmapenlem2 3392  xpmapenlem4 3394  xpmapenlem5 3395

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-dm 2428

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