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Theorem dmsnop 5082
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 5080 . 2  |-  ( B  e.  _V  ->  dom  {
<. A ,  B >. }  =  { A }
)
31, 2ax-mp 5 1  |-  dom  { <. A ,  B >. }  =  { A }
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141   _Vcvv 2730   {csn 3581   <.cop 3584   dom cdm 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-dm 4619
This theorem is referenced by:  dmtpop  5084  dmsnsnsng  5086  op1sta  5090  funtp  5249  ac6sfi  6872
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