Theorem nfunsnafv 47112

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6928. (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 47089	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simprbi 496	. 2 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
3		afvnfundmuv 47109	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
4	2, 3	nsyl5 159	1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3463  {csn 4606  dom cdm 5665   ↾ cres 5667  Fun wfun 6535   defAt wdfat 47086  '''cafv 47087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-res 5677  df-iota 6494  df-fun 6543  df-fv 6549  df-aiota 47055  df-dfat 47089  df-afv 47090

This theorem is referenced by:  afvvfunressn  47113  nfunsnaov  47156