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Theorem afvnufveq 31654
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 31645 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
21con3i 135 . . 3  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  -.  F defAt  A )
3 afvnfundmuv 31646 . . 3  |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
42, 3syl 16 . 2  |-  ( -.  ( F''' A )  =  ( F `  A )  ->  ( F''' A )  =  _V )
54necon1ai 2691 1  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1374    =/= wne 2655   _Vcvv 3106   ` cfv 5579   defAt wdfat 31620  '''cafv 31621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rab 2816  df-v 3108  df-un 3474  df-if 3933  df-fv 5587  df-afv 31624
This theorem is referenced by:  afvvfveq  31655  afvfv0bi  31659  aovnuoveq  31698
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