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Theorem funsn 5029
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 5027 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  Fun  { <. A ,  B >. } )
41, 2, 3mp2an 417 1  |-  Fun  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 1436   _Vcvv 2615   {csn 3431   <.cop 3434   Fun wfun 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-fun 4985
This theorem is referenced by:  funtp  5034  fun0  5039  fvsn  5457
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