Theorem op1sta 4899

Description: Extract the first member of an ordered pair. (See op2nda 4902 to extract the second member and op1stb 4290 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

Step	Hyp	Ref	Expression
1		cnvsn.2	. . . 4 |- B e. _V
2	1	dmsnop 4891	. . 3 |- dom { <. A , B >. } = { A }
3	2	unieqi 3658	. 2 |- U. dom { <. A , B >. } = U. { A }
4		cnvsn.1	. . 3 |- A e. _V
5	4	unisn 3664	. 2 |- U. { A } = A
6	3, 5	eqtri 2108	1 |- U. dom { <. A , B >. } = A

Syntax hints:   = wceq 1289   e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   U.cuni 3648   dom cdm 4428

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-dm 4438

This theorem is referenced by:  op1st  5899  fo1st  5910  f1stres  5912  xpassen  6526  xpdom2  6527