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Theorem afv2ndeffv0 42008
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 3041 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
2 dfatafv2rnb 41975 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 df-dfat 41867 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3bitr3i 268 . . . 4 ((𝐹''''𝐴) ∈ ran 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
54notbii 311 . . 3 (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
6 ianor 1004 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
71, 5, 63bitri 288 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
8 ndmfv 6405 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
9 nfunsn 6413 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
108, 9jaoi 883 . 2 ((¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = ∅)
117, 10sylbi 208 1 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wnel 3040  c0 4079  {csn 4334  dom cdm 5277  ran crn 5278  cres 5279  Fun wfun 6062  cfv 6068   defAt wdfat 41864  ''''cafv2 41956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-dfat 41867  df-afv2 41957
This theorem is referenced by:  afv2fv0b  42014
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