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Theorem afvvfunressn 38515
Description: If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvfunressn  |-  ( ( F''' A )  e.  B  ->  Fun  ( F  |`  { A } ) )

Proof of Theorem afvvfunressn
StepHypRef Expression
1 nfunsnafv 38514 . . 3  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
2 nvelim 38492 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 17 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  ( F''' A )  e.  B
)
43con4i 133 1  |-  ( ( F''' A )  e.  B  ->  Fun  ( F  |`  { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3080   {csn 3998    |` cres 4855   Fun wfun 5595  '''cafv 38486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rab 2780  df-v 3082  df-un 3441  df-if 3912  df-fv 5609  df-dfat 38488  df-afv 38489
This theorem is referenced by:  aovvfunressn  38559
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