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

Theorem afvnufveq 27968
Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvnufveq  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )

Proof of Theorem afvnufveq
StepHypRef Expression
1 afvfundmfveq 27959 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
21con3i 129 . . 3  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  -.  F defAt  A )
3 afvnfundmuv 27960 . . 3  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
42, 3syl 16 . 2  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  ( F''' A )  =  _V )
54necon1ai 2640 1  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    =/= wne 2598   _Vcvv 2948   ` cfv 5446   defAt wdfat 27928  '''cafv 27929
This theorem is referenced by:  afvvfveq  27969  afvfv0bi  27973  aovnuoveq  28012
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-afv 27932
  Copyright terms: Public domain W3C validator