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Theorem afvvv 27678
Description: If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvv  |-  ( ( F''' A )  e.  B  ->  A  e.  _V )

Proof of Theorem afvvv
StepHypRef Expression
1 afvprc 27677 . . 3  |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
2 nvelim 27646 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 16 . 2  |-  ( -.  A  e.  _V  ->  -.  ( F''' A )  e.  B
)
43con4i 124 1  |-  ( ( F''' A )  e.  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899  '''cafv 27640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-un 3268  df-if 3683  df-fv 5402  df-dfat 27642  df-afv 27643
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