Theorem aovrcl 41269

Description: Reverse closure for an operation value, analogous to afvvv 41225. In contrast to ovrcl 6686, 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 41198	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2	1	eleq1i 2692	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
3		afvvdm 41221	. . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
4		df-br 4654	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
5		aovprc.1	. . . . 5 ⊢ Rel dom 𝐹
6		brrelex12 5155	. . . . 5 ⊢ ((Rel dom 𝐹 ∧ 𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	5, 6	mpan 706	. . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	4, 7	sylbir 225	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9	3, 8	syl 17	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
10	2, 9	sylbi 207	1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  Rel wrel 5119  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)