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Theorem fnsn 5445
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 5439 . 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 1717   _Vcvv 2900   {csn 3758   <.cop 3761    Fn wfn 5390
This theorem is referenced by:  f1osn  5656  fvsnun2  5869  elixpsn  7038  axdc3lem4  8267  hashf1lem1  11632  eupath2lem3  21550  cvmliftlem4  24755  cvmliftlem5  24756  axlowdimlem8  25603  axlowdimlem9  25604  axlowdimlem11  25606  axlowdimlem12  25607  bnj927  28478
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-fun 5397  df-fn 5398
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