Theorem dmsnop 5609

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 5606	. 2 |- ( B e. _V -> dom { <. A , B >. } = { A } )
3	1, 2	ax-mp 5	1 |- dom { <. A , B >. } = { A }

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   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-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-br 4654  df-dm 5124

This theorem is referenced by:  dmtpop  5611  dmsnsnsn  5613  op1sta  5617  funtp  5945  funopdmsn  6415  wfrlem13  7427  wfrlem16  7430  tfrlem10  7483  ac6sfi  8204  dcomex  9269  axdc3lem4  9275  cnfldfun  19758  bnj1416  31107  bnj1421  31110  subfacp1lem2a  31162  subfacp1lem5  31166  noextend  31819  nosupbday  31851  nosupbnd1  31860  nosupbnd2  31862