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Theorem ndmafv 27980
Description: The value of a class outside its domain is the universe, compare with ndmfv 5755. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ndmafv  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )

Proof of Theorem ndmafv
StepHypRef Expression
1 df-dfat 27950 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simplbi 447 . . 3  |-  ( F defAt 
A  ->  A  e.  dom  F )
32con3i 129 . 2  |-  ( -.  A  e.  dom  F  ->  -.  F defAt  A )
4 afvnfundmuv 27979 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 16 1  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   dom cdm 4878    |` cres 4880   Fun wfun 5448   defAt wdfat 27947  '''cafv 27948
This theorem is referenced by:  afvvdm  27981  afvprc  27984  afvco2  28016  ndmaov  28023  aovprc  28028
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-un 3325  df-if 3740  df-fv 5462  df-dfat 27950  df-afv 27951
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