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Theorem fnsn 5631
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 5625 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { <. A ,  B >. }  Fn  { A } )
41, 2, 3mp2an 672 1  |-  { <. A ,  B >. }  Fn  { A }
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1804   _Vcvv 3095   {csn 4014   <.cop 4020    Fn wfn 5573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-fun 5580  df-fn 5581
This theorem is referenced by:  f1osn  5843  fnsnb  6075  fvsnun2  6092  elixpsn  7510  axdc3lem4  8836  hashf1lem1  12485  axlowdimlem8  24228  axlowdimlem9  24229  axlowdimlem11  24231  axlowdimlem12  24232  eupath2lem3  24955  cvmliftlem4  28710  cvmliftlem5  28711  finixpnum  30013  bnj927  33560
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