Theorem nfunsnafv 47133

Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6902. (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 47110	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simprbi 496	. 2 ⊢ (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
3		afvnfundmuv 47130	. 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 1540   ∈ wcel 2109  Vcvv 3450  {csn 4591  dom cdm 5640   ↾ cres 5642  Fun wfun 6507   defAt wdfat 47107  '''cafv 47108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-res 5652  df-iota 6466  df-fun 6515  df-fv 6521  df-aiota 47076  df-dfat 47110  df-afv 47111

This theorem is referenced by:  afvvfunressn  47134  nfunsnaov  47177