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Theorem afvnfundmuv 37573
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 37566 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iffalse 3893 . 2  |-  ( -.  F defAt  A  ->  if ( F defAt  A ,  ( F `  A ) ,  _V )  =  _V )
31, 2syl5eq 2455 1  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1405   _Vcvv 3058   ifcif 3884   ` cfv 5568   defAt wdfat 37547  '''cafv 37548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-un 3418  df-if 3885  df-fv 5576  df-afv 37551
This theorem is referenced by:  ndmafv  37574  nfunsnafv  37576  afvnufveq  37581  afvres  37606  afvco2  37610  aovnfundmuv  37616
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