Theorem afvnfundmuv 41219

Description: If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
afvnfundmuv	|- ( -. F defAt A -> ( F''' A ) = _V )

Proof of Theorem afvnfundmuv

Step	Hyp	Ref	Expression
1		dfafv2 41212	. 2 |- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )
2		iffalse 4095	. 2 |- ( -. F defAt A -> if ( F defAt A , ( F ` A ) , _V ) = _V )
3	1, 2	syl5eq 2668	1 |- ( -. F defAt A -> ( F''' A ) = _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   _Vcvv 3200   ifcif 4086   ` cfv 5888   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-afv 41197

This theorem is referenced by:  ndmafv  41220  nfunsnafv  41222  afvnufveq  41227  afvres  41252  afvco2  41256  aovnfundmuv  41262