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Theorem nfunsnafv 38043
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5912 (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 38017 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simprbi 465 . . 3  |-  ( F defAt 
A  ->  Fun  ( F  |`  { A } ) )
32con3i 140 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  F defAt  A )
4 afvnfundmuv 38040 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 17 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002   dom cdm 4854    |` cres 4856   Fun wfun 5595   defAt wdfat 38014  '''cafv 38015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-un 3447  df-if 3916  df-fv 5609  df-dfat 38017  df-afv 38018
This theorem is referenced by:  afvvfunressn  38044  nfunsnaov  38087
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