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Theorem afv0fv0 41733
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 4940 . . 3 ∅ ∈ V
2 eleq1a 2832 . . 3 (∅ ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V))
31, 2ax-mp 5 . 2 ((𝐹'''𝐴) = ∅ → (𝐹'''𝐴) ∈ V)
4 afvvfveq 41732 . . 3 ((𝐹'''𝐴) ∈ V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2762 . . . 4 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65biimpd 219 . . 3 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅))
74, 6syl 17 . 2 ((𝐹'''𝐴) ∈ V → ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅))
83, 7mpcom 38 1 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2137  Vcvv 3338  c0 4056  cfv 6047  '''cafv 41698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-nul 4057  df-if 4229  df-fv 6055  df-afv 41701
This theorem is referenced by:  afvfv0bi  41736  aov0ov0  41777
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