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Theorem dmsnop 5344
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 5341 . 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 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817   dom cdm 4878
This theorem is referenced by:  dmtpop  5346  dmsnsnsn  5348  op1sta  5351  funtp  5503  tfrlem10  6648  ac6sfi  7351  dcomex  8327  axdc3lem4  8333  wlkntrllem1  21559  eupap1  21698  ablosn  21935  subfacp1lem2a  24866  subfacp1lem5  24870  wfrlem13  25550  wfrlem16  25553  bnj1416  29408  bnj1421  29411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-dm 4888
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