Theorem aovprc 47814

Description: The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7449. (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 47747	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		df-br 5114	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3		aovprc.1	. . . . 5 ⊢ Rel dom 𝐹
4	3	brrelex12i 5717	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	2, 4	sylbir 238	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		ndmafv 47766	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹'''⟨𝐴, 𝐵⟩) = V)
7	5, 6	nsyl5 160	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹'''⟨𝐴, 𝐵⟩) = V)
8	1, 7	eqtrid 2816	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   class class class wbr 5113  dom cdm 5662  Rel wrel 5667  '''cafv 47743   ((caov 47744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47711  df-dfat 47745  df-afv 47746  df-aov 47747

This theorem is referenced by: (None)