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Theorem ndmafv2nrn 47172
Description: The value of a class outside its domain is not in the range, compare with ndmfv 6942. (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 867 . . 3 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2 ianor 983 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 47069 . . . 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 47171 . 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 847  wcel 2106  wnel 3044  {csn 4631  dom cdm 5689  ran crn 5690  cres 5691  Fun wfun 6557   defAt wdfat 47066  ''''cafv2 47158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-dfat 47069  df-afv2 47159
This theorem is referenced by:  afv2prc  47176  fafv2elrnb  47185
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