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Theorem funsn 5485
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 5483 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  Fun  { <. A ,  B >. } )
41, 2, 3mp2an 654 1  |-  Fun  { <. A ,  B >. }
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2943   {csn 3801   <.cop 3804   Fun wfun 5434
This theorem is referenced by:  funtp  5489  fun0  5494  fvsn  5912  dcomex  8311  axdc3lem4  8317  xpsc0  13768  xpsc1  13769  wfrlem13  25518  bnj1421  29163
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 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
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-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-fun 5442
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