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

Theorem nfunsnafv 27996
Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5764 (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
nfunsnafv  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )

Proof of Theorem nfunsnafv
StepHypRef Expression
1 df-dfat 27964 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
21simprbi 452 . . 3  |-  ( F defAt 
A  ->  Fun  ( F  |`  { A } ) )
32con3i 130 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  F defAt  A )
4 afvnfundmuv 27993 . 2  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
53, 4syl 16 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F''' A )  =  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   dom cdm 4881    |` cres 4883   Fun wfun 5451   defAt wdfat 27961  '''cafv 27962
This theorem is referenced by:  afvvfunressn  27997  nfunsnaov  28040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-un 3327  df-if 3742  df-fv 5465  df-dfat 27964  df-afv 27965
  Copyright terms: Public domain W3C validator