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

Theorem afvvfveq 31716
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
afvvfveq  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )

Proof of Theorem afvvfveq
StepHypRef Expression
1 nvelim 31688 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
21necon2ai 2702 . 2  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =/=  _V )
3 afvnufveq 31715 . 2  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3syl 16 1  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   ` cfv 5587  '''cafv 31682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-un 3481  df-if 3940  df-fv 5595  df-afv 31685
This theorem is referenced by:  afv0fv0  31717  afv0nbfvbi  31719  aovvoveq  31760
  Copyright terms: Public domain W3C validator