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Theorem afvvdm 38456
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 38455 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
2 nvelim 38435 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 17 . 2  |-  ( -.  A  e.  dom  F  ->  -.  ( F''' A )  e.  B )
43con4i 133 1  |-  ( ( F''' A )  e.  B  ->  A  e.  dom  F
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3022   dom cdm 4796  '''cafv 38429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rab 2723  df-v 3024  df-un 3384  df-if 3855  df-fv 5552  df-dfat 38431  df-afv 38432
This theorem is referenced by:  aovvdm  38500  aovrcl  38504  aoprssdm  38517
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