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Theorem afv2ndeffv0 47272
Description: If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv2ndeffv0 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)

Proof of Theorem afv2ndeffv0
StepHypRef Expression
1 df-nel 3047 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
2 dfatafv2rnb 47239 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 df-dfat 47131 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3bitr3i 277 . . . 4 ((𝐹''''𝐴) ∈ ran 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
54notbii 320 . . 3 (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
6 ianor 984 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
71, 5, 63bitri 297 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
8 ndmfv 6941 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
9 nfunsn 6948 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
108, 9jaoi 858 . 2 ((¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = ∅)
117, 10sylbi 217 1 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wnel 3046  c0 4333  {csn 4626  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555  cfv 6561   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  afv2fv0b  47278
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