Theorem dmsnop 4872

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 4870	. 2 |- ( B e. _V -> dom { <. A , B >. } = { A } )
3	1, 2	ax-mp 7	1 |- dom { <. A , B >. } = { A }

Syntax hints:   = wceq 1287   e. wcel 1436   _Vcvv 2615   {csn 3431   <.cop 3434   dom cdm 4413

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-dm 4423

This theorem is referenced by:  dmtpop  4874  dmsnsnsng  4876  op1sta  4880  funtp  5034  ac6sfi  6568