Theorem nfunsnaov 47663

Description: If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
nfunsnaov	⊢ (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)

Proof of Theorem nfunsnaov

Step	Hyp	Ref	Expression
1		df-aov 47598	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		nfunsnafv 47619	. 2 ⊢ (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → (𝐹'''⟨𝐴, 𝐵⟩) = V)
3	1, 2	eqtrid 2788	1 ⊢ (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1548  Vcvv 3433  {csn 4558  ⟨cop 4564   ↾ cres 5623  Fun wfun 6483  '''cafv 47594   ((caov 47595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497  df-aiota 47562  df-dfat 47596  df-afv 47597  df-aov 47598

This theorem is referenced by: (None)