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Theorem afv20fv0 45427
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 45396 . 2 ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2 dfatafv2eqfv 45425 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
32eqcomd 2742 . . . 4 (𝐹 defAt 𝐴 → (𝐹𝐴) = (𝐹''''𝐴))
43adantr 481 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = (𝐹''''𝐴))
5 simpr 485 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
64, 5eqtrd 2776 . 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 396   = wceq 1541  c0 4280  cfv 6493   defAt wdfat 45280  ''''cafv2 45372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-fv 6501  df-afv2 45373
This theorem is referenced by:  afv2fv0b  45430
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