Theorem nfunsnafv 41222

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6225. (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 41196	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simprbi 480	. . 3 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
3	2	con3i 150	. 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
4		afvnfundmuv 41219	. 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 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   defAt wdfat 41193  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  afvvfunressn  41223  nfunsnaov  41266