Theorem afvvdm 41221

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

Step	Hyp	Ref	Expression
1		ndmafv 41220	. . 3 |- ( -. A e. dom F -> ( F''' A ) = _V )
2		nvelim 41200	. . 3 |- ( ( F''' A ) = _V -> -. ( F''' A ) e. B )
3	1, 2	syl 17	. 2 |- ( -. A e. dom F -> -. ( F''' A ) e. B )
4	3	con4i 113	1 |- ( ( F''' A ) e. B -> A e. dom F )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   dom cdm 5114  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  aovvdm  41265  aovrcl  41269  aoprssdm  41282