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Theorem nfunsnafv 27873
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5720 (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
nfunsnafv  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )

Proof of Theorem nfunsnafv
StepHypRef Expression
1 df-dfat 27841 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simprbi 451 . . 3  |-  ( F defAt 
A  ->  Fun  ( F  |`  { A } ) )
32con3i 129 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  F defAt  A )
4 afvnfundmuv 27870 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 16 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774   dom cdm 4837    |` cres 4839   Fun wfun 5407   defAt wdfat 27838  '''cafv 27839
This theorem is referenced by:  afvvfunressn  27874  nfunsnaov  27917
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-un 3285  df-if 3700  df-fv 5421  df-dfat 27841  df-afv 27842
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