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Theorem afv20defat 42128
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 42107 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwne0 5059 . . . . 5 𝒫 ran 𝐹 ≠ ∅
32neii 3001 . . . 4 ¬ 𝒫 ran 𝐹 = ∅
4 eqeq1 2829 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ran 𝐹 = ∅))
53, 4mtbiri 319 . . 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 1656  c0 4146  𝒫 cpw 4380   cuni 4660  ran crn 5347   defAt wdfat 42012  ''''cafv2 42104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-afv2 42105
This theorem is referenced by:  afv20fv0  42159
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