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Theorem ndmafv 27153
Description: The value of a class outside its domain is the universe, compare with ndmfv 5590. (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 27122 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simplbi 446 . . 3  |-  ( F defAt 
A  ->  A  e.  dom  F )
32con3i 127 . 2  |-  ( -.  A  e.  dom  F  ->  -.  F defAt  A )
4 afvnfundmuv 27152 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 15 1  |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1633    e. wcel 1701   _Vcvv 2822   {csn 3674   dom cdm 4726    |` cres 4728   Fun wfun 5286   defAt wdfat 27119  '''cafv 27120
This theorem is referenced by:  afvvdm  27154  afvprc  27157  afvco2  27189  ndmaov  27196  aovprc  27201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-un 3191  df-if 3600  df-fv 5300  df-dfat 27122  df-afv 27123
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