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Theorem fnsng 5381
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fnsng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )

Proof of Theorem fnsng
StepHypRef Expression
1 funsng 5380 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )
2 dmsnopg 5226 . . 3  |-  ( B  e.  W  ->  dom  {
<. A ,  B >. }  =  { A }
)
32adantl 452 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  dom  { <. A ,  B >. }  =  { A } )
4 df-fn 5340 . 2  |-  ( {
<. A ,  B >. }  Fn  { A }  <->  ( Fun  { <. A ,  B >. }  /\  dom  {
<. A ,  B >. }  =  { A }
) )
51, 3, 4sylanbrc 645 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {csn 3716   <.cop 3719   dom cdm 4771   Fun wfun 5331    Fn wfn 5332
This theorem is referenced by:  fnsn  5386  fnunsn  5433  fsnunfv  5804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-fun 5339  df-fn 5340
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