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Theorem aovvfunressn 28029
Description: If the operation 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, 26-May-2017.)
Assertion
Ref Expression
aovvfunressn  |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } ) )

Proof of Theorem aovvfunressn
StepHypRef Expression
1 df-aov 27954 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2501 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfunressn 27985 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } ) )
42, 3sylbi 189 1  |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   {csn 3816   <.cop 3819    |` cres 4882   Fun wfun 5450  '''cafv 27950   ((caov 27951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-un 3327  df-if 3742  df-fv 5464  df-dfat 27952  df-afv 27953  df-aov 27954
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