Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afv2ndeffv0 Structured version   Visualization version   GIF version

Theorem afv2ndeffv0 43811
 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 3092 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
2 dfatafv2rnb 43778 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 df-dfat 43670 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3bitr3i 280 . . . 4 ((𝐹''''𝐴) ∈ ran 𝐹 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
54notbii 323 . . 3 (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
6 ianor 979 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
71, 5, 63bitri 300 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
8 ndmfv 6675 . . 3 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
9 nfunsn 6682 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
108, 9jaoi 854 . 2 ((¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = ∅)
117, 10sylbi 220 1 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091  ∅c0 4243  {csn 4525  dom cdm 5519  ran crn 5520   ↾ cres 5521  Fun wfun 6318  ‘cfv 6324   defAt wdfat 43667  ''''cafv2 43759 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-dfat 43670  df-afv2 43760 This theorem is referenced by:  afv2fv0b  43817
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