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Theorem afv20defat 46517
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 46496 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwne0 5348 . . . . 5 𝒫 ran 𝐹 ≠ ∅
32neii 2936 . . . 4 ¬ 𝒫 ran 𝐹 = ∅
4 eqeq1 2730 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ran 𝐹 = ∅))
53, 4mtbiri 327 . . 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 1533  c0 4317  𝒫 cpw 4597   cuni 4902  ran crn 5670   defAt wdfat 46401  ''''cafv2 46493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-afv2 46494
This theorem is referenced by:  afv20fv0  46548
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