Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvnufveq Structured version   Visualization version   GIF version

Theorem afvnufveq 41548
 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 41539 . . . 4 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
21con3i 150 . . 3 (¬ (𝐹'''𝐴) = (𝐹𝐴) → ¬ 𝐹 defAt 𝐴)
3 afvnfundmuv 41540 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
42, 3syl 17 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
54necon1ai 2850 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ≠ wne 2823  Vcvv 3231  ‘cfv 5926   defAt wdfat 41514  '''cafv 41515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rab 2950  df-v 3233  df-un 3612  df-if 4120  df-fv 5934  df-afv 41518 This theorem is referenced by:  afvvfveq  41549  afvfv0bi  41553  aovnuoveq  41592
 Copyright terms: Public domain W3C validator