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Theorem ndmafv2nrn 47234
Description: The value of a class outside its domain is not in the range, compare with ndmfv 6941. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndmafv2nrn 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndmafv2nrn
StepHypRef Expression
1 orc 868 . . 3 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2 ianor 984 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 47131 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3xchnxbir 333 . . 3 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
51, 4sylibr 234 . 2 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴)
6 ndfatafv2nrn 47233 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
75, 6syl 17 1 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  wcel 2108  wnel 3046  {csn 4626  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   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-ext 2708  ax-sep 5296  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-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  afv2prc  47238  fafv2elrnb  47247
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