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Theorem afv20defat 45538
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 45517 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwne0 5317 . . . . 5 𝒫 ran 𝐹 ≠ ∅
32neii 2946 . . . 4 ¬ 𝒫 ran 𝐹 = ∅
4 eqeq1 2741 . . . 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 1542  c0 4287  𝒫 cpw 4565   cuni 4870  ran crn 5639   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-v 3450  df-dif 3918  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-afv2 45515
This theorem is referenced by:  afv20fv0  45569
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