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Theorem afvnfundmuv 31648
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 31641 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iffalse 3943 . 2  |-  ( -.  F defAt  A  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  _V )
31, 2syl5eq 2515 1  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1374   _Vcvv 3108   ifcif 3934   ` cfv 5581   defAt wdfat 31622  '''cafv 31623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-un 3476  df-if 3935  df-fv 5589  df-afv 31626
This theorem is referenced by:  ndmafv  31649  nfunsnafv  31651  afvnufveq  31656  afvres  31681  afvco2  31685  aovnfundmuv  31691
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