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Theorem afvnufveq 27680
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 27671 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
21con3i 129 . . 3  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  -.  F defAt  A )
3 afvnfundmuv 27672 . . 3  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
42, 3syl 16 . 2  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  ( F''' A )  =  _V )
54necon1ai 2592 1  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    =/= wne 2550   _Vcvv 2899   ` cfv 5394   defAt wdfat 27639  '''cafv 27640
This theorem is referenced by:  afvvfveq  27681  afvfv0bi  27685  aovnuoveq  27724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-un 3268  df-if 3683  df-fv 5402  df-afv 27643
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