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Theorem afv20defat 43727
 Description: If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv20defat ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)

Proof of Theorem afv20defat
StepHypRef Expression
1 ndfatafv2 43706 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwne0 5234 . . . . 5 𝒫 ran 𝐹 ≠ ∅
32neii 3013 . . . 4 ¬ 𝒫 ran 𝐹 = ∅
4 eqeq1 2826 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ran 𝐹 = ∅))
53, 4mtbiri 330 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ¬ (𝐹''''𝐴) = ∅)
61, 5syl 17 . 2 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) = ∅)
76con4i 114 1 ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538  ∅c0 4265  𝒫 cpw 4511  ∪ cuni 4813  ran crn 5533   defAt wdfat 43611  ''''cafv2 43703 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-afv2 43704 This theorem is referenced by:  afv20fv0  43758
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