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Theorem afvvv 28113
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 28112 . . 3  |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
2 nvelim 28081 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 15 . 2  |-  ( -.  A  e.  _V  ->  -.  ( F''' A )  e.  B
)
43con4i 122 1  |-  ( ( F''' A )  e.  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801  '''cafv 28075
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
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