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Theorem afvnufveq 40990
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) = (𝐹𝐴))

Proof of Theorem afvnufveq
StepHypRef Expression
1 afvfundmfveq 40981 . . . 4 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
21con3i 150 . . 3 (¬ (𝐹'''𝐴) = (𝐹𝐴) → ¬ 𝐹 defAt 𝐴)
3 afvnfundmuv 40982 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
42, 3syl 17 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
54necon1ai 2818 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wne 2791  Vcvv 3195  cfv 5876   defAt wdfat 40956  '''cafv 40957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-rab 2918  df-v 3197  df-un 3572  df-if 4078  df-fv 5884  df-afv 40960
This theorem is referenced by:  afvvfveq  40991  afvfv0bi  40995  aovnuoveq  41034
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