Theorem ndmafv 41220

Description: The value of a class outside its domain is the universe, compare with ndmfv 6218. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
ndmafv	|- ( -. A e. dom F -> ( F''' A ) = _V )

Proof of Theorem ndmafv

Step	Hyp	Ref	Expression
1		df-dfat 41196	. . . 4 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
2	1	simplbi 476	. . 3 |- ( F defAt A -> A e. dom F )
3	2	con3i 150	. 2 |- ( -. A e. dom F -> -. F defAt A )
4		afvnfundmuv 41219	. 2 |- ( -. F defAt A -> ( F''' A ) = _V )
5	3, 4	syl 17	1 |- ( -. A e. dom F -> ( F''' A ) = _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   dom cdm 5114   |` cres 5116   Fun wfun 5882   defAt wdfat 41193  '''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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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:  afvvdm  41221  afvprc  41224  afvco2  41256  ndmaov  41263  aovprc  41268