Theorem dmsnop 5082

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 5080	. 2 |- ( B e. _V -> dom { <. A , B >. } = { A } )
3	1, 2	ax-mp 5	1 |- dom { <. A , B >. } = { A }

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   {csn 3581   <.cop 3584   dom cdm 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-dm 4619

This theorem is referenced by:  dmtpop  5084  dmsnsnsng  5086  op1sta  5090  funtp  5249  ac6sfi  6872