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Theorem afv0fv0 39878
 Description: If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afv0fv0 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)

Proof of Theorem afv0fv0
StepHypRef Expression
1 0ex 4718 . . 3 ∅ ∈ V
2 eleq1a 2683 . . 3 (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V))
31, 2ax-mp 5 . 2 ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)
4 afvvfveq 39877 . . 3 ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2614 . . . 4 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65biimpd 218 . . 3 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅))
74, 6syl 17 . 2 ((𝐹'''𝐴) ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅))
83, 7mpcom 37 1 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  '''cafv 39843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-if 4037  df-fv 5812  df-afv 39846 This theorem is referenced by:  afvfv0bi  39881  aov0ov0  39922
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