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Theorem afv20fv0 46876
Description: If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv20fv0 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)

Proof of Theorem afv20fv0
StepHypRef Expression
1 afv20defat 46845 . 2 ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2 dfatafv2eqfv 46874 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
32eqcomd 2732 . . . 4 (𝐹 defAt 𝐴 → (𝐹𝐴) = (𝐹''''𝐴))
43adantr 479 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = (𝐹''''𝐴))
5 simpr 483 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
64, 5eqtrd 2766 . 2 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = ∅)
71, 6mpancom 686 1 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  c0 4325  cfv 6554   defAt wdfat 46729  ''''cafv2 46821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-fv 6562  df-afv2 46822
This theorem is referenced by:  afv2fv0b  46879
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