Theorem nfunsnafv 47899

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6920. (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 47876	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simprbi 502	. 2 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
3		afvnfundmuv 47896	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
4	2, 3	nsyl5 160	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589  dom cdm 5661   ↾ cres 5663  Fun wfun 6530   defAt wdfat 47873  '''cafv 47874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-aiota 47842  df-dfat 47876  df-afv 47877

This theorem is referenced by:  afvvfunressn  47900  nfunsnaov  47943