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Theorem afvnufveq 41754
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 41745 . . . 4 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
21con3i 151 . . 3 (¬ (𝐹'''𝐴) = (𝐹𝐴) → ¬ 𝐹 defAt 𝐴)
3 afvnfundmuv 41746 . . 3 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
42, 3syl 17 . 2 (¬ (𝐹'''𝐴) = (𝐹𝐴) → (𝐹'''𝐴) = V)
54necon1ai 3016 1 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1637  wne 2989  Vcvv 3402  cfv 6111   defAt wdfat 41723  '''cafv 41724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-br 4856  df-opab 4918  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-res 5336  df-iota 6074  df-fun 6113  df-fv 6119  df-aiota 41687  df-dfat 41726  df-afv 41727
This theorem is referenced by:  afvvfveq  41755  afvfv0bi  41759  aovnuoveq  41798
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