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Theorem afvnufveq 28115
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 28106 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
21con3i 127 . . 3  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  -.  F defAt  A )
3 afvnfundmuv 28107 . . 3  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
42, 3syl 15 . 2  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  ( F''' A )  =  _V )
54necon1ai 2501 1  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    =/= wne 2459   _Vcvv 2801   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  afvvfveq  28116  afvfv0bi  28120  aovnuoveq  28159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-fv 5279  df-afv 28078
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