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Theorem fnsn 5320
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 5315 . 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 1696   _Vcvv 2801   {csn 3653   <.cop 3656    Fn wfn 5266
This theorem is referenced by:  f1osn  5529  fvsnun2  5732  elixpsn  6871  axdc3lem4  8095  hashf1lem1  11409  cvmliftlem4  23834  cvmliftlem5  23835  eupath2lem3  23918  axlowdimlem8  24649  axlowdimlem9  24650  axlowdimlem11  24652  axlowdimlem12  24653  bnj927  29116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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