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Theorem fnsn 5304
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 5299 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { <. A ,  B >. }  Fn  { A } )
41, 2, 3mp2an 653 1  |-  { <. A ,  B >. }  Fn  { A }
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    Fn wfn 5250
This theorem is referenced by:  f1osn  5513  fvsnun2  5716  elixpsn  6855  axdc3lem4  8079  hashf1lem1  11393  cvmliftlem4  23819  cvmliftlem5  23820  eupath2lem3  23903  axlowdimlem8  24577  axlowdimlem9  24578  axlowdimlem11  24580  axlowdimlem12  24581  bnj927  28800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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