Theorem nfunsnafv2 47846

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 6918. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
nfunsnafv2	⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem nfunsnafv2

Step	Hyp	Ref	Expression
1		olc 881	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2		ianor 997	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47740	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	xchnxbir 336	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
5	1, 4	sylibr 237	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
6		ndfatafv2nrn 47842	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
7	5, 6	syl 18	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∈ wcel 2149   ∉ wnel 3070  {csn 4591  dom cdm 5659  ran crn 5660   ↾ cres 5661  Fun wfun 6528   defAt wdfat 47737  ''''cafv2 47829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nel 3071  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-if 4490  df-pw 4566  df-uni 4874  df-dfat 47740  df-afv2 47830

This theorem is referenced by: (None)