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Theorem aovprc 43742
Description: The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7173. (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
StepHypRef Expression
1 df-aov 43675 . 2 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2 df-br 5031 . . . 4 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
3 aovprc.1 . . . . 5 Rel dom 𝐹
43brrelex12i 5571 . . . 4 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
52, 4sylbir 238 . . 3 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6 ndmafv 43694 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹'''⟨𝐴, 𝐵⟩) = V)
75, 6nsyl5 162 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹'''⟨𝐴, 𝐵⟩) = V)
81, 7syl5eq 2845 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  cop 4531   class class class wbr 5030  dom cdm 5519  Rel wrel 5524  '''cafv 43671   ((caov 43672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-aiota 43640  df-dfat 43673  df-afv 43674  df-aov 43675
This theorem is referenced by: (None)
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