Theorem nfunsnafv2 45133

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 6872. (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 866	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2		ianor 980	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 45027	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	xchnxbir 333	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
5	1, 4	sylibr 233	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
6		ndfatafv2nrn 45129	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
7	5, 6	syl 17	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 845   ∈ wcel 2106   ∉ wnel 3047  {csn 4578  dom cdm 5625  ran crn 5626   ↾ cres 5627  Fun wfun 6478   defAt wdfat 45024  ''''cafv2 45116

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5248  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nel 3048  df-rab 3405  df-v 3444  df-un 3907  df-in 3909  df-ss 3919  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-uni 4858  df-dfat 45027  df-afv2 45117

This theorem is referenced by: (None)