Theorem aovvfunressn 41267

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

Step	Hyp	Ref	Expression
1		df-aov 41198	. . 3 |- (( A F B)) = ( F''' <. A , B >. )
2	1	eleq1i 2692	. 2 |- ( (( A F B)) e. C <-> ( F''' <. A , B >. ) e. C )
3		afvvfunressn 41223	. 2 |- ( ( F''' <. A , B >. ) e. C -> Fun ( F |` { <. A , B >. } ) )
4	2, 3	sylbi 207	1 |- ( (( A F B)) e. C -> Fun ( F |` { <. A , B >. } ) )

Syntax hints:   -> wi 4   e. wcel 1990   {csn 4177   <.cop 4183   |` cres 5116   Fun wfun 5882  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)