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Theorem afv20fv0 45569
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 45538 . 2 ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2 dfatafv2eqfv 45567 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
32eqcomd 2743 . . . 4 (𝐹 defAt 𝐴 → (𝐹𝐴) = (𝐹''''𝐴))
43adantr 482 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = (𝐹''''𝐴))
5 simpr 486 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
64, 5eqtrd 2777 . 2 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = ∅)
71, 6mpancom 687 1 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  c0 4287  cfv 6501   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5268
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-fv 6509  df-afv2 45515
This theorem is referenced by:  afv2fv0b  45572
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