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Theorem fnsn 5496
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
fnsn.1  |-  A  e. 
_V
fnsn.2  |-  B  e. 
_V
Assertion
Ref Expression
fnsn  |-  { <. A ,  B >. }  Fn  { A }

Proof of Theorem fnsn
StepHypRef Expression
1 fnsn.1 . 2  |-  A  e. 
_V
2 fnsn.2 . 2  |-  B  e. 
_V
3 fnsng 5490 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { <. A ,  B >. }  Fn  { A } )
41, 2, 3mp2an 654 1  |-  { <. A ,  B >. }  Fn  { A }
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809    Fn wfn 5441
This theorem is referenced by:  f1osn  5707  fvsnun2  5921  elixpsn  7093  axdc3lem4  8323  hashf1lem1  11694  eupath2lem3  21691  cvmliftlem4  24965  cvmliftlem5  24966  axlowdimlem8  25853  axlowdimlem9  25854  axlowdimlem11  25856  axlowdimlem12  25857  bnj927  29040
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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