Theorem nfunsnafv2 47779

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 6900. (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 879	. . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2		ianor 994	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47673	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	xchnxbir 335	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
5	1, 4	sylibr 236	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
6		ndfatafv2nrn 47775	. 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 399   ∨ wo 858   ∈ wcel 2141   ∉ wnel 3060  {csn 4579  dom cdm 5643  ran crn 5644   ↾ cres 5645  Fun wfun 6509   defAt wdfat 47670  ''''cafv2 47762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nel 3061  df-rab 3414  df-v 3455  df-in 3909  df-ss 3919  df-if 4478  df-pw 4554  df-uni 4863  df-dfat 47673  df-afv2 47763

This theorem is referenced by: (None)