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Theorem afvvfunressn 27156
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 27155 . . 3  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
2 nvelim 27126 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 15 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  ( F''' A )  e.  B
)
43con4i 122 1  |-  ( ( F''' A )  e.  B  ->  Fun  ( F  |`  { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1633    e. wcel 1701   _Vcvv 2822   {csn 3674    |` cres 4728   Fun wfun 5286  '''cafv 27120
This theorem is referenced by:  aovvfunressn  27200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-un 3191  df-if 3600  df-fv 5300  df-dfat 27122  df-afv 27123
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