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Theorem afvvdm 32429
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 32428 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
2 nvelim 32408 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
31, 2syl 16 . 2  |-  ( -.  A  e.  dom  F  ->  -.  ( F''' A )  e.  B )
43con4i 130 1  |-  ( ( F''' A )  e.  B  ->  A  e.  dom  F
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   dom cdm 5008  '''cafv 32402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-un 3476  df-if 3945  df-fv 5602  df-dfat 32404  df-afv 32405
This theorem is referenced by:  aovvdm  32473  aovrcl  32477  aoprssdm  32490
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