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Theorem afvnufveq 42889
 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 42880 . . . 4 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
21con3i 157 . . 3 (¬ (𝐹'''𝐴) = (𝐹𝐴) → ¬ 𝐹 defAt 𝐴)
3 afvnfundmuv 42881 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
42, 3syl 17 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
54necon1ai 3011 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1522   ≠ wne 2984  Vcvv 3437  ‘cfv 6230   defAt wdfat 42858  '''cafv 42859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-int 4787  df-br 4967  df-opab 5029  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-res 5460  df-iota 6194  df-fun 6232  df-fv 6238  df-aiota 42828  df-dfat 42861  df-afv 42862 This theorem is referenced by:  afvvfveq  42890  afvfv0bi  42894  aovnuoveq  42933
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