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Theorem nfunsnafv 27330
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5641 (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 27297 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simprbi 450 . . 3  |-  ( F defAt 
A  ->  Fun  ( F  |`  { A } ) )
32con3i 127 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  F defAt  A )
4 afvnfundmuv 27327 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 15 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1642    e. wcel 1710   _Vcvv 2864   {csn 3716   dom cdm 4771    |` cres 4773   Fun wfun 5331   defAt wdfat 27294  '''cafv 27295
This theorem is referenced by:  afvvfunressn  27331  nfunsnaov  27374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-un 3233  df-if 3642  df-fv 5345  df-dfat 27297  df-afv 27298
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