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Theorem afvvv 27976
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 27975 . . 3  |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
2 nvelim 27945 . . 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 1652    e. wcel 1725   _Vcvv 2948  '''cafv 27939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-dfat 27941  df-afv 27942
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