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Theorem nfunsnafv 30188
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5822 (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 30160 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simprbi 464 . . 3  |-  ( F defAt 
A  ->  Fun  ( F  |`  { A } ) )
32con3i 135 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  F defAt  A )
4 afvnfundmuv 30185 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 16 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070   {csn 3977   dom cdm 4940    |` cres 4942   Fun wfun 5512   defAt wdfat 30157  '''cafv 30158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3072  df-un 3433  df-if 3892  df-fv 5526  df-dfat 30160  df-afv 30161
This theorem is referenced by:  afvvfunressn  30189  nfunsnaov  30232
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