Theorem op1sta 5250

Description: Extract the first member of an ordered pair. (See op2nda 5253 to extract the second member and op1stb 4605 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 5242	. . 3 |- dom { <. A , B >. } = { A }
3	2	unieqi 3930	. 2 |- U. dom { <. A , B >. } = U. { A }
4		cnvsn.1	. . 3 |- A e. _V
5	4	unisn 3936	. 2 |- U. { A } = A
6	3, 5	eqtri 2255	1 |- U. dom { <. A , B >. } = A

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3695   <.cop 3698   U.cuni 3920   dom cdm 4755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-dm 4765

This theorem is referenced by:  op1st  6354  fo1st  6365  f1stres  6367  xpassen  7095  xpdom2  7096