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Theorem afvvfveq 38520
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 38492 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
21necon2ai 2655 . 2  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =/=  _V )
3 afvnufveq 38519 . 2  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3syl 17 1  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080   ` cfv 5601  '''cafv 38486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rab 2780  df-v 3082  df-un 3441  df-if 3912  df-fv 5609  df-afv 38489
This theorem is referenced by:  afv0fv0  38521  afv0nbfvbi  38523  aovvoveq  38564
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