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Theorem afvvdm 27983
Description: If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvdm  |-  ( ( F''' A )  e.  B  ->  A  e.  dom  F
)

Proof of Theorem afvvdm
StepHypRef Expression
1 ndmafv 27982 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
2 nvelim 27956 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 16 . 2  |-  ( -.  A  e.  dom  F  ->  -.  ( F''' A )  e.  B )
43con4i 125 1  |-  ( ( F''' A )  e.  B  ->  A  e.  dom  F
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   dom cdm 4880  '''cafv 27950
This theorem is referenced by:  aovvdm  28027  aovrcl  28031  aoprssdm  28044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-un 3327  df-if 3742  df-fv 5464  df-dfat 27952  df-afv 27953
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