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Theorem aovrcl 39918
 Description: Reverse closure for an operation value, analogous to afvvv 39874. In contrast to ovrcl 6584, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aovprc.1 Rel dom 𝐹
Assertion
Ref Expression
aovrcl ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem aovrcl
StepHypRef Expression
1 df-aov 39847 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21eleq1i 2679 . 2 ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3 afvvdm 39870 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
4 df-br 4584 . . . 4 (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
5 aovprc.1 . . . . 5 Rel dom 𝐹
6 brrelex12 5079 . . . . 5 ((Rel dom 𝐹𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
75, 6mpan 702 . . . 4 (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
84, 7sylbir 224 . . 3 (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
93, 8syl 17 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
102, 9sylbi 206 1 ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  Rel wrel 5043  '''cafv 39843   ((caov 39844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847 This theorem is referenced by: (None)
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