Theorem aovvfunressn 47165

Description: If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovvfunressn	⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))

Proof of Theorem aovvfunressn

Step	Hyp	Ref	Expression
1		df-aov 47099	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	eleq1i 2832	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3		afvvfunressn 47121	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))
4	2, 3	sylbi 217	1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))

Syntax hints:   → wi 4   ∈ wcel 2108  {csn 4634  ⟨cop 4640   ↾ cres 5695  Fun wfun 6563  '''cafv 47095   ((caov 47096

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-res 5705  df-iota 6522  df-fun 6571  df-fv 6577  df-aiota 47063  df-dfat 47097  df-afv 47098  df-aov 47099

This theorem is referenced by: (None)