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Theorem funsn 5283
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
Hypotheses
Ref Expression
funsn.1  |-  A  e. 
_V
funsn.2  |-  B  e. 
_V
Assertion
Ref Expression
funsn  |-  Fun  { <. A ,  B >. }

Proof of Theorem funsn
StepHypRef Expression
1 funsn.1 . 2  |-  A  e. 
_V
2 funsn.2 . 2  |-  B  e. 
_V
3 funsng 5281 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  Fun  { <. A ,  B >. } )
41, 2, 3mp2an 426 1  |-  Fun  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 2160   _Vcvv 2752   {csn 3607   <.cop 3610   Fun wfun 5229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-fun 5237
This theorem is referenced by:  funtp  5288  fun0  5293  fvsn  5731
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