Theorem aovvfunressn 47297

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 47231	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	eleq1i 2822	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3		afvvfunressn 47253	. 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 2111  {csn 4573  ⟨cop 4579   ↾ cres 5616  Fun wfun 6475  '''cafv 47227   ((caov 47228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-aiota 47195  df-dfat 47229  df-afv 47230  df-aov 47231

This theorem is referenced by: (None)