Theorem dmsnop 5077

Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
dmsnop.1	|- B e. _V

Assertion

Ref	Expression
dmsnop	|- dom { <. A , B >. } = { A }

Proof of Theorem dmsnop

Step	Hyp	Ref	Expression
1		dmsnop.1	. 2 |- B e. _V
2		dmsnopg 5075	. 2 |- ( B e. _V -> dom { <. A , B >. } = { A } )
3	1, 2	ax-mp 5	1 |- dom { <. A , B >. } = { A }

Syntax hints:   = wceq 1343   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   dom cdm 4604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-dm 4614

This theorem is referenced by:  dmtpop  5079  dmsnsnsng  5081  op1sta  5085  funtp  5241  ac6sfi  6864