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Theorem afvnfundmuv 27672
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
StepHypRef Expression
1 dfafv2 27665 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iffalse 3689 . 2  |-  ( -.  F defAt  A  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  _V )
31, 2syl5eq 2431 1  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649   _Vcvv 2899   ifcif 3682   ` cfv 5394   defAt wdfat 27639  '''cafv 27640
This theorem is referenced by:  ndmafv  27673  nfunsnafv  27675  afvnufveq  27680  afvres  27705  afvco2  27709  aovnfundmuv  27715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-un 3268  df-if 3683  df-fv 5402  df-afv 27643
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