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Theorem dmsnop 5473
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 5470 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
31, 2ax-mp 5 1  |-  dom  { <. A ,  B >. }  =  { A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   _Vcvv 3106   {csn 4020   <.cop 4026   dom cdm 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-dm 5002
This theorem is referenced by:  dmtpop  5475  dmsnsnsn  5477  op1sta  5481  funtp  5631  tfrlem10  7046  ac6sfi  7753  dcomex  8816  axdc3lem4  8822  wlkntrllem1  24223  eupap1  24638  ablosn  25011  subfacp1lem2a  28250  subfacp1lem5  28254  wfrlem13  28918  wfrlem16  28921  bnj1416  33049  bnj1421  33052
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