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Theorem nfunsnafv2 45531
Description: If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6889. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
nfunsnafv2 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem nfunsnafv2
StepHypRef Expression
1 olc 867 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2 ianor 981 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 45425 . . . 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 233 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
6 ndfatafv2nrn 45527 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
75, 6syl 17 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  wcel 2107  wnel 3050  {csn 4591  dom cdm 5638  ran crn 5639  cres 5640  Fun wfun 6495   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nel 3051  df-rab 3411  df-v 3450  df-un 3920  df-in 3922  df-ss 3932  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4871  df-dfat 45425  df-afv2 45515
This theorem is referenced by: (None)
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