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Theorem afv20fv0 42297
 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 42266 . 2 ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
2 dfatafv2eqfv 42295 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
32eqcomd 2783 . . . 4 (𝐹 defAt 𝐴 → (𝐹𝐴) = (𝐹''''𝐴))
43adantr 474 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = (𝐹''''𝐴))
5 simpr 479 . . 3 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹''''𝐴) = ∅)
64, 5eqtrd 2813 . 2 ((𝐹 defAt 𝐴 ∧ (𝐹''''𝐴) = ∅) → (𝐹𝐴) = ∅)
71, 6mpancom 678 1 ((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  ∅c0 4140  ‘cfv 6135   defAt wdfat 42150  ''''cafv2 42242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-fv 6143  df-afv2 42243 This theorem is referenced by:  afv2fv0b  42300
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