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Theorem fnsng 5465
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 5464 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  Fun  { <. A ,  B >. } )
2 dmsnopg 5308 . . 3  |-  ( B  e.  W  ->  dom  {
<. A ,  B >. }  =  { A }
)
32adantl 453 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  dom  { <. A ,  B >. }  =  { A } )
4 df-fn 5424 . 2  |-  ( {
<. A ,  B >. }  Fn  { A }  <->  ( Fun  { <. A ,  B >. }  /\  dom  {
<. A ,  B >. }  =  { A }
) )
51, 3, 4sylanbrc 646 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3782   <.cop 3785   dom cdm 4845   Fun wfun 5415    Fn wfn 5416
This theorem is referenced by:  fnsn  5471  fnunsn  5519  fsnunfv  5900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-fun 5423  df-fn 5424
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