Theorem aovrcl 44681

Description: Reverse closure for an operation value, analogous to afvvv 44637. In contrast to ovrcl 7316, 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

Step	Hyp	Ref	Expression
1		df-aov 44613	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	eleq1i 2829	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3		afvvdm 44633	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
4		df-br 5075	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
5		aovprc.1	. . . . 5 ⊢ Rel dom 𝐹
6	5	brrelex12i 5642	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	4, 6	sylbir 234	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	3, 7	syl 17	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	2, 8	sylbi 216	1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  Rel wrel 5594  '''cafv 44609   ((caov 44610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612  df-aov 44613

This theorem is referenced by: (None)