Theorem op1sta 5617

Description: Extract the first member of an ordered pair. (See op2nda 5620 to extract the second member, op1stb 4940 for an alternate version, and op1st 7176 for the preferred 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 5609	. . 3 |- dom { <. A , B >. } = { A }
3	2	unieqi 4445	. 2 |- U. dom { <. A , B >. } = U. { A }
4		cnvsn.1	. . 3 |- A e. _V
5	4	unisn 4451	. 2 |- U. { A } = A
6	3, 5	eqtri 2644	1 |- U. dom { <. A , B >. } = A

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U.cuni 4436   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124

This theorem is referenced by:  elxp4  7110  op1st  7176  fo1st  7188  f1stres  7190  xpassen  8054  xpdom2  8055