Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aovvfunressn Unicode version

Theorem aovvfunressn 28155
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 28079 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2359 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfunressn 28111 . 2  |-  ( ( F''' <. A ,  B >. )  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } ) )
42, 3sylbi 187 1  |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   {csn 3653   <.cop 3656    |` cres 4707   Fun wfun 5265  '''cafv 28075   ((caov 28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-fv 5279  df-dfat 28077  df-afv 28078  df-aov 28079
  Copyright terms: Public domain W3C validator