Theorem op1sta 5164

Description: Extract the first member of an ordered pair. (See op2nda 5167 to extract the second member and op1stb 4525 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 5156	. . 3 |- dom { <. A , B >. } = { A }
3	2	unieqi 3860	. 2 |- U. dom { <. A , B >. } = U. { A }
4		cnvsn.1	. . 3 |- A e. _V
5	4	unisn 3866	. 2 |- U. { A } = A
6	3, 5	eqtri 2226	1 |- U. dom { <. A , B >. } = A

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   <.cop 3636   U.cuni 3850   dom cdm 4675

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-dm 4685

This theorem is referenced by:  op1st  6232  fo1st  6243  f1stres  6245  xpassen  6925  xpdom2  6926