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Theorem afvnufveq 45532
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 45523 . . 3 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2 afvnfundmuv 45524 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
31, 2nsyl5 159 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
43necon1ai 2967 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wne 2939  Vcvv 3459  cfv 6516   defAt wdfat 45501  '''cafv 45502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-br 5126  df-opab 5188  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-res 5665  df-iota 6468  df-fun 6518  df-fv 6524  df-aiota 45470  df-dfat 45504  df-afv 45505
This theorem is referenced by:  afvvfveq  45533  afvfv0bi  45537  aovnuoveq  45576
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