Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndmafv Structured version   Unicode version

Theorem ndmafv 32464
Description: The value of a class outside its domain is the universe, compare with ndmfv 5872. (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 32440 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simplbi 458 . . 3  |-  ( F defAt 
A  ->  A  e.  dom  F )
32con3i 135 . 2  |-  ( -.  A  e.  dom  F  ->  -.  F defAt  A )
4 afvnfundmuv 32463 . 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 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   dom cdm 4988    |` cres 4990   Fun wfun 5564   defAt wdfat 32437  '''cafv 32438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-un 3466  df-if 3930  df-fv 5578  df-dfat 32440  df-afv 32441
This theorem is referenced by:  afvvdm  32465  afvprc  32468  afvco2  32500  ndmaov  32507  aovprc  32512
  Copyright terms: Public domain W3C validator