Theorem nfunsnafv 39871

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6135. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
nfunsnafv	⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Proof of Theorem nfunsnafv

Step	Hyp	Ref	Expression
1		df-dfat 39845	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simprbi 479	. . 3 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
3	2	con3i 149	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
4		afvnfundmuv 39868	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
5	3, 4	syl 17	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   defAt wdfat 39842  '''cafv 39843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037  df-fv 5812  df-dfat 39845  df-afv 39846

This theorem is referenced by:  afvvfunressn  39872  nfunsnaov  39915