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Theorem dmsnop 5147
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
StepHypRef Expression
1 dmsnop.1 . 2  |-  B  e. 
_V
2 dmsnopg 5144 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
31, 2ax-mp 8 1  |-  dom  { <. A ,  B >. }  =  { A }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   dom cdm 4689
This theorem is referenced by:  dmtpop  5149  dmsnsnsn  5151  op1sta  5154  funtp  5303  tfrlem10  6403  ac6sfi  7101  dcomex  8073  axdc3lem4  8079  ablosn  21014  subfacp1lem2a  23711  subfacp1lem5  23715  eupap1  23900  wfrlem13  24268  wfrlem16  24271  1alg  25722  1ded  25738  1cat  25759  bnj1416  29069  bnj1421  29072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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