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Theorem afvnfundmuv 27970
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 27963 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iffalse 3738 . 2  |-  ( -.  F defAt  A  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  _V )
31, 2syl5eq 2479 1  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652   _Vcvv 2948   ifcif 3731   ` cfv 5446   defAt wdfat 27938  '''cafv 27939
This theorem is referenced by:  ndmafv  27971  nfunsnafv  27973  afvnufveq  27978  afvres  28003  afvco2  28007  aovnfundmuv  28013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-afv 27942
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