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Theorem aovvfunressn 28047
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 27976 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
21eleq1i 2346 . 2  |-  ( (( A F B))  e.  C  <->  ( F''' <. A ,  B >. )  e.  C )
3 afvvfunressn 28006 . 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 1684   {csn 3640   <.cop 3643    |` cres 4691   Fun wfun 5249  '''cafv 27972   ((caov 27973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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