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Theorem ndmafv 30189
Description: The value of a class outside its domain is the universe, compare with ndmfv 5818. (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 30163 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simplbi 460 . . 3  |-  ( F defAt 
A  ->  A  e.  dom  F )
32con3i 135 . 2  |-  ( -.  A  e.  dom  F  ->  -.  F defAt  A )
4 afvnfundmuv 30188 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 16 1  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3072   {csn 3980   dom cdm 4943    |` cres 4945   Fun wfun 5515   defAt wdfat 30160  '''cafv 30161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-rab 2805  df-v 3074  df-un 3436  df-if 3895  df-fv 5529  df-dfat 30163  df-afv 30164
This theorem is referenced by:  afvvdm  30190  afvprc  30193  afvco2  30225  ndmaov  30232  aovprc  30237
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