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Theorem dmsnop 5465
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 5462 . 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 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   <.cop 4022   dom cdm 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-dm 4998
This theorem is referenced by:  dmtpop  5467  dmsnsnsn  5469  op1sta  5473  funtp  5622  tfrlem10  7048  ac6sfi  7756  dcomex  8818  axdc3lem4  8824  wlkntrllem1  24763  eupap1  25178  ablosn  25547  subfacp1lem2a  28888  subfacp1lem5  28892  wfrlem13  29595  wfrlem16  29598  bnj1416  34496  bnj1421  34499
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