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Theorem afvvfveq 27989
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 27955 . . 3  |-  ( ( F''' A )  =  _V  ->  -.  ( F''' A )  e.  B )
21necon2ai 2650 . 2  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =/=  _V )
3 afvnufveq 27988 . 2  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3syl 16 1  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2600   _Vcvv 2957   ` cfv 5455  '''cafv 27949
This theorem is referenced by:  afv0fv0  27990  afv0nbfvbi  27992  aovvoveq  28033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-un 3326  df-if 3741  df-fv 5463  df-afv 27952
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