Theorem aovprc 43394

Description: The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7196. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aovprc.1	⊢ Rel dom 𝐹

Assertion

Ref	Expression
aovprc	⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Proof of Theorem aovprc

Step	Hyp	Ref	Expression
1		df-aov 43327	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		df-br 5069	. . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3		aovprc.1	. . . . . 6 ⊢ Rel dom 𝐹
4	3	brrelex12i 5609	. . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	2, 4	sylbir 237	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	5	con3i 157	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
7		ndmafv 43346	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹'''⟨𝐴, 𝐵⟩) = V)
8	6, 7	syl 17	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹'''⟨𝐴, 𝐵⟩) = V)
9	1, 8	syl5eq 2870	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   class class class wbr 5068  dom cdm 5557  Rel wrel 5562  '''cafv 43323   ((caov 43324

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365  df-aiota 43292  df-dfat 43325  df-afv 43326  df-aov 43327

This theorem is referenced by: (None)